Adversarial examination of the union-closed-family bounds

Subject and independence

Examination date: 2026-10-15 UTC. Subject frozen: 2026-10-15 at 13:17:10.128162 UTC, as recorded in the supplied neutral manifest. Reviewer: independent mathematical examiner, disclosed model gpt-6-astra. The reviewer neither authored nor collaborated on the subject and received no earlier review or author working context. The examination was charged to find a counterexample, an unsupported essential inference, or a convention failure.

The subject is the complete $F/F_k$ content of Theorems 1.1 and 1.2 and Corollary 1.3(i)--(iv) of Deep Bhattacharjee, Priyabrata Mandal and Ushashi Bhattacharya, Resolving Erdos-Ulam Monochromatic Union-Closed Family Conjectures, arXiv:2610.02833v1, original Erdos.tex. The source's theorem labels and mathematical credits remain theirs. The neutral model and packet assembly are Plasma AI work; the independent derivations in this report are also Plasma AI work. The Gowers method credit and the Sauer and Shelah citations are retained below.

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This is a text-based whole-proof mathematical examination, with the explicit Sauer--Shelah premise stated below. It is not original-PDF fidelity, TeX rendering, kernel validation, source installation, source acceptance, or an examination of the excluded sublattice quantity $f$. The other source versions named in the provenance file were not read or substituted.

Model and quantified restatements

For every positive integer $n$, write $[n]=\{1,\ldots,n\}$. For an integer $k\ge2$, a coloring is any map $\chi:2^{[n]}\to[k]$, with unused colors allowed. A family $\mathcal A\subseteq2^{[n]}$ is union-closed if $A\cup B\in\mathcal A$ for every $A,B\in\mathcal A$. There is no intersection requirement and no requirement that $\varnothing\in\mathcal A$. Put

$$F_k(n)=\min_\chi\max\{|\mathcal A|: \mathcal A\text{ is monochromatic and union-closed}\}, \qquad F(n)=F_2(n).$$

Both extrema are attained over finite collections. A singleton family is always available, so $1\le F_k(n)\le2^n$. Embedding the two colors in $[k]$ shows $F_k(n)\le F(n)$: the maximum for each embedded coloring is unchanged, and the minimum is then taken over a larger set of colorings. These statements concern cardinalities of actual families, not the number of their generators.

Theorem 1.1: polynomial bound

For every pair of integers $k\ge2$, $t\ge1$, define

$$M=t^2k^{2^t}, \qquad c_{k,t}=\bigl(k^{2^t+1}t^t(4M^2)^{t+1}\bigr)^{-1}.$$

For every integer $n\ge1$ and every $k$-coloring of all subsets of $[n]$, there is a monochromatic union-closed family of at least $c_{k,t}n^{t+1}$ members. Equivalently, $F_k(n)\ge c_{k,t}n^{t+1}$. The constant depends on both $k$ and $t$, not on $n$ or the coloring.

Theorem 1.1: two-color threshold

For every integer $t\ge1$ and positive integer $n$ with $\log_2 n\ge(t+2)^2 2^t$, one has $F(n)\ge n^t$. The non-strict threshold includes equality.

Theorem 1.2: binomial bound

For every positive integer $n$ and integer $d\ge1$ satisfying the strict hypothesis $2^d>nd+2$,

$$F(n)\le\sum_{j=0}^{d-1}\binom nj.$$

Here $\binom nj=0$ for $j>n$. There is no hypothesis $d\le n$.

Theorem 1.2: explicit choice

For every integer $n\ge2$, the integer

$$d=\lfloor\log_2 n\rfloor +\lfloor\log_2\lfloor\log_2 n\rfloor\rfloor+3$$

satisfies the strict hypothesis above and gives

$$F(n)\le d\,n^{d-1} \le(\log_2 n+\log_2\log_2 n+3) n^{\log_2 n+\log_2\log_2 n+2}.$$

Corollary 1.3(i)

For every integer $n\ge2$, with $\ell=\log_2\log_2 n$ and $\omega(n)=\ell-2\log_2(\ell+2)-1$, $F(n)\ge n^{\omega(n)}$, and $\omega(n)\to\infty$ as $n\to\infty$. Negative values of the displayed exponent are permitted.

Corollary 1.3(ii)

There exists a real-valued function $\varepsilon$ on the positive integers such that $\varepsilon(n)\to0$ and $F(n)<(1+\varepsilon(n))^n$ for every positive integer $n$. The asserted inequality is strict at every $n$, including $n=1$.

Corollary 1.3(iii)

For every real-valued function $\omega$ on the positive integers such that $F(n)\ge n^{\omega(n)}$ for every positive integer $n$, and every real $\varepsilon>0$, there exists $N$ such that $\omega(n)\le(1+\varepsilon)\log_2 n$ for every integer $n\ge N$. No monotonicity or divergence assumption on this $\omega$ is made.

Corollary 1.3(iv)

For each fixed integer $k\ge2$, there exist real-valued functions $\omega_k,\varepsilon_k$ on the positive integers with $\omega_k(n)\to\infty$ and $\varepsilon_k(n)\to0$, such that

$$n^{\omega_k(n)}\le F_k(n)<(1+\varepsilon_k(n))^n \quad\text{for every integer }n\ge1.$$

The lower-growth assertion is for each fixed $k$; it does not assert uniform divergence when $k$ is allowed to grow with $n$.

Three weakest steps, independently rederived

The three derivations below cover the delicate internal interfaces rather than treating same-paper lemmas as accepted premises. The consequence checks following them give the remaining deductions and the strongest attacks for each assigned conclusion.

1. From the box inequality to one usable cube shape

The consumed text is box_and_supersaturation.tex.txt, lines 10--141, together with the cube and shape definitions in cubes_and_gluing.tex.txt, lines 4--72.

A cube with disjoint base $U$ and nonempty disjoint generators $X_1,\ldots,X_t$ has $2^t$ different vertices: membership of any point of $X_i$ recovers whether generator $i$ was selected. Union of two vertices corresponds to union of their index sets; intersection corresponds to intersection of those index sets. The top-down description follows by complementing the index set inside $[t]$. This also handles $t=0$, when the cube is the singleton $\{U\}$. The intersection identity is used only to understand this cube, not to strengthen the assigned family model.

Two cubes of the same ordered shape are carried to one another by a permutation: biject corresponding base, generator and unused parts. Right composition by that permutation bijects the permutations producing a monochromatic image. Thus $\rho_\chi(u;a_1,\ldots,a_t)$ is independent of the chosen representative. Uniform permutations give uniform images on their orbit, since fibers have equal size. Equal generator sizes or different representations of the same cube do not affect this argument. The proof never assumes that different shapes are equally likely.

Here is an independent induction for the box inequality, using products of two slices rather than the source's first application of Cauchy--Schwarz. Write $\mathcal B_t(f)$ for the box average of a nonnegative function. For $t=0$ it is the value at the sole point of $G^0$. For $t\ge1$, take $a,a'\in G$, $y\in G^{t-1}$ and put $q=2^{t-1}$. Applying the induction hypothesis to the nonnegative function $y\mapsto f(a,y)f(a',y)$ gives

$$\begin{aligned} \mathcal B_t(f) &=\mathbb E_{a,a'}\mathcal B_{t-1} \bigl(y\mapsto f(a,y)f(a',y)\bigr)\\ &\ge\mathbb E_{a,a'} \bigl(\mathbb E_y f(a,y)f(a',y)\bigr)^q\\ &\ge\bigl(\mathbb E_y(\mathbb E_a f(a,y))^2\bigr)^q\\ &\ge(\mathbb E f)^{2q}. \end{aligned}$$

The second inequality is Jensen for the exponent $q\ge1$, after exchanging finite sums and using independence of $a,a'$. The last is Cauchy--Schwarz applied to the mean over $y$. All quantities are nonnegative, so raising them to $q$ preserves inequalities. This proves the exponent $2^t$ without an assumption about the desired conclusion. The source's induction also works: its first squared mean is bounded by Cauchy--Schwarz, and its subsequent slice bound and Jensen inequality have the same nonnegative domains. For $t=1$, $q=1$ makes the relevant Jensen step an equality.

A uniform distribution on $G$ presupposes $G\ne\varnothing$ when $t>0$. The application uses $G=\{0,\ldots,L\}$ with $L\ge1$, so it satisfies this domain condition. No assertion about a uniform random point of an empty set is needed.

For fixed $\sigma$, let $p_c$ be the density of color $c$ on this grid. The box inequality applied to each color indicator and Jensen give

$$\Pr(E\mid\sigma)\ge\sum_{c=1}^k p_c^{2^t} \ge k^{1-2^t}.$$

The monochromatic events for different colors are disjoint even for degenerate boxes. The event $D$ that some coordinate pair coincides has probability at most $t/(L+1)$, by a union bound. Therefore

$$\Pr(E\setminus D)\ge k^{1-2^t}-\frac{t}{L+1}.$$

For a pair outside $D$, put $\ell_i=\min(x_i,x'_i)$ and $h_i=\max(x_i,x'_i)$. The sets at its vertices have base $U=S(\ell)$ and generators consisting of positions $\ell_i+1,\ldots,h_i$ in chain $Z_i$. Disjoint chains make these generators disjoint from each other and from the base. Their shape satisfies

$$u=r+\sum_i\ell_i,\qquad a_i=h_i-\ell_i\ge1,\qquad (u-r)+\sum_i a_i=\sum_i h_i\le tL=M.$$

There is no conditioning error when a single shape is chosen. For each nondegenerate pair its conditional probability over $\sigma$ is exactly the corresponding $\rho_\chi$. Writing its nonnegative sampling weight as $w_\eta$,

$$\Pr(E\setminus D)=\sum_{\eta\notin D}w_\eta\rho_\chi(\eta) \le \max_{\eta\notin D}\rho_\chi(\eta).$$

The nondegenerate collection is nonempty because $L\ge1$; for instance, the all-zero and all-one grid points form such a pair. The weights sum to at most one. Thus the maximum is a value for an admissible shape, including when the displayed general lower bound is zero or negative. No shape uniformity, and no shape independent of the given coloring, is asserted.

For $L=t k^{2^t}$ and $M=tL$, the degeneracy loss satisfies $t/(L+1)<k^{-2^t}$. The surviving lower bound is strictly larger than $(k-1)k^{-2^t}$, hence at least $k^{-2^t}$ for every $k\ge2$. Also $u-r\le M-t$ and every $a_i\le M$. These are exactly the interfaces used by the slices argument. The numerical examples $M_2(1)=4$, $M_2(2)=64$, $M_2(3)=2304$ follow directly, and $M\ge4t^2\ge t+1$ for all allowed $k,t$.

This derivation discharges the internal box and supersaturation obligations. The source's description of a box norm and its Gowers attribution supply method context; no norm theorem or result from Gowers's paper is imported.

2. From separately good slices to one large union-closed family

The consumed text is lower_proof.tex.txt, lines 4--109, and the block-chain, union-rule and gluing arguments in cubes_and_gluing.tex.txt, lines 74--155. The main attack is that existence of a good shape for each slice need not give one block vector good for many pairs. The exact incidence count below supplies that missing implication; it does not assume simultaneous choices.

Let $m=\lfloor(n-1)/M\rfloor$ and $s=\lfloor m/2\rfloor$, initially in the regime $s\ge t$. Then $m\ge2$, and $m$ disjoint slots of size $M$ fit inside $[n]$. Every vector $\beta\in[M]^m$ specifies nonempty disjoint blocks. For $A\subseteq[j-1]$ with $|A|=t$, freeze the coordinates outside $A\cup\{j\}$ and put

$$R=\sum_{\substack{i<j\\i\notin A}}\beta_i,\qquad r=R+1.$$

There are exactly $m-t-1$ frozen coordinates. Since $j\ge t+1$,

$$0\le R\le(j-1-t)M\le(m-2)M, \qquad r+M\le(m-1)M+1\le n.$$

Supersaturation therefore supplies a shape $(u;a_1,\ldots,a_t)$ with $u-r=z\in[0,M-t]$ and $\rho_\chi\ge k^{-2^t}$. Taking $\beta_j=z+1$ and assigning $a_1,\ldots,a_t$ in increasing order to the indices in $A$ gives entries in $[M]$ and exactly the desired shape: its base size is $R+\beta_j=u$. In particular, the $+1$ both makes the level block nonempty and matches the base. This works at $j=t+1$, where $R=0$, as well as at later levels.

Every fixed assignment extends, and different frozen assignments yield different full vectors. Consequently there are at least $M^{m-t-1}$ good vectors for each pair. This lower bound does not say that their sets of extensions have a large intersection.

Set $\mathcal P=\{(j,A):s<j\le m,\ A\subseteq[s],\ |A|=t\}$. For each good vector at one pair, at least $n!k^{-2^t}$ permutations make its cube monochromatic. Counting incidences between pairs $(\sigma,\beta)$ and elements of $\mathcal P$ gives at least

$$|\mathcal P|\,M^{m-t-1}\,n!k^{-2^t}$$

incidences among $n!M^m$ pairs $(\sigma,\beta)$. Hence at least one common pair has at least $|\mathcal P|k^{-2^t}M^{-(t+1)}$ monochromatic cubes. No independence between these cube events is used. A real-valued lower bound on an integer count is legitimate, with the integer automatically at least its ceiling.

It remains to turn this cube count into cardinality of a union-closed family. Write $H(j,B)=C_j\setminus Y_B$. For $j\le j'$ and $B,B'\subseteq[j]$, a point in a block with index at most $j$ is missing from the union precisely when both factors omit it; blocks with indices between $j+1$ and $j'$ are all present. Thus

$$H(j,B)\cup H(j',B')=H(j',B\cap B').$$

For the cubes being glued, $B,B'\subseteq[s]$ and $j>s$, so this identity applies. Choose a color possessed by at least $1/k$ of the selected monochromatic cubes. Their union as families is monochromatic. The union of two of its members lies in the cube at the larger level: $B\cap B'\subseteq B'\subseteq A'$. This proves closure in the actual union-family model without requiring intersections or an empty member.

Cube overlap cannot erase the needed cardinality. The base $H(j,A)$ contains the whole nonempty block $Y_j$, contains no later block, and omits exactly the blocks with indices in $A$ among $[s]$. It therefore determines $j$ and $A$. These bases inject the chosen pairs into the glued family, giving at least

$$\frac{|\mathcal P|}{k^{2^t+1}M^{t+1}}$$

distinct members. An empty selection would make the gluing conclusion trivial; in the large-parameter application the positive incidence bound already yields a nonempty selection.

For $n\ge4M(t+1)$, the floor estimates are valid including at equality. Because $n$ and $M(m+1)$ are integers, $n-1<M(m+1)$ implies $n\le M(m+1)\le2M(s+1)$. Thus $s\ge2t+1$, which establishes every needed condition $s\ge t$ and $1\le s<m$. Also $n\le4Ms$ and $m-s\ge s$. For $s\ge t$ each factor $(s-i)/(t-i)$, $0\le i<t$, is at least $s/t$, because $i(s-t)\ge0$. It follows that

$$|\mathcal P|=(m-s)\binom st \ge \frac{s^{t+1}}{t^t} \ge\frac{n^{t+1}}{t^t(4M)^{t+1}}.$$

Combining the two denominators gives $M^{t+1}(4M)^{t+1}=(4M^2)^{t+1}$, with exactly the stated $k^{2^t+1}$ factor. This proves the polynomial bound in the large regime. For $1\le n<4M(t+1)\le4M^2$,

$$c_{k,t}n^{t+1} =\frac{(n/(4M^2))^{t+1}}{k^{2^t+1}t^t}<1\le F_k(n).$$

Thus no value of $n$ is lost when slots and slices are unavailable. At the cutoff itself, $m=4t+3$ and $s=2t+1$, so the large-regime argument begins without a floor ambiguity.

For the two-color threshold, expanding the constant gives exactly

$$\log_2(c_{2,t}^{-1}) =(2t+3)2^t+(5t+4)\log_2 t+2t+3.$$

For all integers $t\ge1$, the elementary induction $2^{t-1}\ge t$ implies both $\log_2 t\le t-1$ and $2^t\ge2t$. The remainder is bounded by

$$\begin{aligned} (5t+4)\log_2 t+2t+3 &\le5t^2+t-1\\ &\le2t(t+1)^2\\ &\le(t+1)^2 2^t. \end{aligned}$$

For the middle inequality, the difference is $2t^3-t^2+t+1=t^2(2t-1)+t+1>0$. Since $(2t+3)+(t+1)^2=(t+2)^2$, this proves $\log_2(c_{2,t}^{-1})\le(t+2)^2 2^t$ for every allowed $t$. In particular, $t=1$ gives $c_{2,1}=2^{-15}$ and a permitted threshold $\log_2 n\ge18$; it is not an omitted exception. At any allowed threshold $c_{2,t}n\ge1$, so the already established all-$n$ polynomial bound implies $F(n)\ge n^t$.

3. From nonempty unions to the probabilistic upper bound

The consumed text is upper_proof.tex.txt, lines 4--60. Its sole external step is Lemma 5.1(c), with the exact Sauer--Shelah interface recorded separately below.

If $(A_1,\ldots,A_d)$ is an independent system and $x_i$ is private to $A_i$, the $x_i$ are distinct. The trace of $\bigcup_{i\in I}A_i$ on $X=\{x_1,\ldots,x_d\}$ is exactly $\{x_i:i\in I\}$. Therefore its $2^d-1$ nonempty-index unions are distinct. The sets $A_i$ may overlap outside $X$; pairwise disjointness is not used.

If a family shatters a $d$-element set $X$, choosing a member with trace $\{x_i\}$ for each $i$ gives just such a private-element system. Shattering is inherited by subsets, so absence of an independent system of size $d$ rules out all shattered sets of size at least $d$. The auxiliary $d=0$ statements cause no difficulty: there are no nonempty index unions to distinguish and an empty system is available. The random-coloring theorem itself only uses $d\ge1$.

For fixed $d\ge1$, let every subset of $[n]$ receive an independent fair bit. A fixed independent tuple is bad exactly when its $N=2^d-1$ distinct nonempty unions all get one bit. As $N\ge1$, the two possible constant assignments are disjoint, and

$$\Pr(\text{this tuple is bad})=2\cdot2^{-N}=2^{2-2^d}.$$

This is where the missing empty union matters: using $2^d$ vertices would give the wrong constant. There are at most $2^{nd}$ ordered tuples. Non-independent tuples contribute no bad event. A union bound, which needs no independence between different tuples, therefore gives

$$\Pr(\text{some bad tuple})\le2^{nd+2-2^d}<1.$$

The strict hypothesis yields a coloring with no bad tuple. Fix that single coloring. In any monochromatic union-closed family, binary closure implies closure under each nonempty finite union by induction on the number of sets being joined. An independent $d$-tuple in that family would consequently be bad. Hence every nonempty such family has VC dimension at most $d-1$, and Sauer--Shelah with $q=d-1\ge0$ bounds its cardinality by $\sum_{j<d}\binom nj$.

An empty family has cardinality zero and can be disposed of directly. The printed definition of VC dimension as a largest shattered size does not specify a value for the empty family; this proof does not need one. For nonempty families the empty test set is shattered, so the definition has a finite maximum. The minimax maximum is always attained by a nonempty family. This explicit treatment avoids turning an unstated convention into a premise.

The resulting bound is on every actual monochromatic union-closed family of the selected coloring, so taking its maximum and then the minimum over colorings gives exactly the asserted bound on $F$. If $d>n$, no independent $d$-system exists and the binomial sum is $2^n$, so this case is also valid directly. For positive $n$, the hypotheses cannot hold at $d=1$ or $d=2$; the theorem does not silently use either case. At $n=1$, they first hold at $d=3$, again with the trivial full binomial sum. The auxiliary cases therefore create no unsupported all-positive-$n$ assertion.

For the explicit choice, let $a=\lfloor\log_2 n\rfloor\ge1$ and $b=\lfloor\log_2 a\rfloor\ge0$. Integrality strengthens the strict power inequalities to $2^{a+1}\ge n+1$ and $2^{b+1}\ge a+1$. The inequality $2^{a-1}\ge a$ gives $b+1\le a$. For $d=a+b+3$,

$$\begin{aligned} 2^d&=2\cdot2^{a+1}\cdot2^{b+1} \ge2(n+1)(a+1),\\ nd+2&=n(a+1)+n(b+2)+2 \le2n(a+1)+2. \end{aligned}$$

The former lower bound exceeds the latter upper bound by $2a>0$. This checks the strict hypothesis at every floor transition, not just asymptotically. In particular, $n=2,3$ give $d=4$; at $n=3$ the comparison still has positive margin $16-14=2$.

For $j\ge0$, $\binom nj\le n^j$, including $j=0$ and the zero terms $j>n$. Thus $\sum_{j<d}\binom nj\le d n^{d-1}$. If $\lambda=\log_2 n\ge1$ and $D=\lambda+\log_2\lambda+3$, then $d\le D$; positivity and $n\ge2$ give $d n^{d-1}\le D n^{D-1}$. Both factors are compared in the required direction. No estimate involving $\log_2\log_2 n$ is applied at $n=1$.

Conclusion-specific attacks, composition and verdicts

The checklist sections below apply to each conclusion in this section. Each conclusion has its own restatement above, its consumed interfaces and weakest-step reference here, and its own verdict. No favorable verdict is inferred merely from another conclusion's name or from an author's output.

Theorem 1.1: polynomial bound

Strongest attack. Good shapes could depend on the frozen coordinates in incompatible ways, and even a large number of same-colored cubes could overlap almost completely. These are two successive ways that a count of local witnesses might fail to give a large actual family. Derivation 2 reconstructs the incidence count over all $(\sigma,\beta,(j,A))$ and the injective map $(j,A)\mapsto H(j,A)$. It thereby defeats both objections: the former selects one common pair, while the latter still distinguishes all selected bases even when many cubes share their top.

As an additional adversarial coloring, take $\chi(A)=|A|\pmod k$. A cube is monochromatic exactly when every generator size is divisible by $k$. Thus a lower bound that silently treated arbitrary shapes as good would fail. The actual argument needs only one good extension per slice. Its free generator coordinates can all be $k$, since $k\le M$, so this coloring supplies no obstruction. The proof's averaging over shapes makes no prohibited uniform-goodness assertion.

The universal $k,n$ quantifiers also survive the extreme case $k\ge2^n$, where an injective coloring gives $F_k(n)=1$. Here $M\ge k^2\ge4^n>n$, so the source's small-$n$ branch gives $c_{k,t}n^{t+1}<1$. It does not demand an impossible large family.

Premises and composition. Derivations 1 and 2 verify all internal cube, permutation, box, supersaturation, slice, averaging, gluing and constant interfaces. Only the stated elementary finite and inequality principles are used; Sauer--Shelah is not a premise for this conclusion. The small-$n$ and large-$n$ branches partition all positive integers.

Verdict: refutation-failed. The exact polynomial bound, including its constant and every integer $k\ge2$, $t\ge1$, $n\ge1$, survives.

Theorem 1.1: two-color threshold

Strongest attack. Recomputing $c_{2,t}$ could expose a lost factor of $M$, or a failure of the logarithmic bound at small $t$, leaving $c_{2,t}n<1$ at the advertised threshold. Derivation 2 recomputes both $M$ factors and the full logarithm, and proves the needed comparison for every $t\ge1$. The displayed positive polynomial difference checks the potentially weak estimate without relying on a finite sample. The endpoint $t=1$ is explicitly covered, and threshold equality still gives $c_{2,t}n\ge1$.

Premises and composition. This conclusion consumes the polynomial bound already examined in this report, at $k=2$, and the final constant calculation of Derivation 2. There is no new external theorem premise.

Verdict: refutation-failed. The stated non-strict threshold implies $F(n)\ge n^t$ with the claimed quantifiers.

Theorem 1.2: binomial bound

Strongest attack. A union-closed family can omit the empty set, and sets obtained from shattering can share a large common part. These facts could invalidate a random-coloring argument written for disjoint generators or a full $2^d$-vertex cube. The following exact test exposes both distinctions. On $[d]\cup\{p\}$, consider

$$\mathcal H=\{\{p\}\cup I:I\subseteq[d]\}, \qquad A_i=\{p,i\}.$$

This is a union-closed family omitting $\varnothing$, it shatters $[d]$, and its chosen generators all overlap at $p$. Nonetheless $i$ is private to $A_i$, and their nonempty unions are exactly the $2^d-1$ sets $\{p\}\cup I$ with $I\ne\varnothing$. The source's probability $2^{2-2^d}$ is the correct one for these sets. Its argument does not use either the empty union or disjointness of the $A_i$.

Derivation 3 also checks the dependence between overlapping bad events. It uses only a union bound, so that dependence cannot invalidate the estimate. The strict exponent $nd+2-2^d<0$ gives positive probability of no bad event. The proof does not infer this from a non-strict probability bound of one.

Premises and composition. The private-element and shattering implications, nonempty finite-union closure, tuple count, probability and minimax transfer are verified in Derivation 3. Sauer--Shelah is used exactly once, with $q=d-1$, as the explicit classical external premise; its original proof is not examined. The empty-family case and $d>n$ are handled there directly.

Verdict: refutation-failed, relying on the stated Sauer--Shelah premise. Every allowed $n,d$ in the conditional binomial assertion is covered.

Theorem 1.2: explicit choice

Strongest attack. Taking $n$ just below a power of two, and $a$ just below a power of two, can minimize the slack in the floor estimates. At the smallest allowed $a=1$, logarithms and the inequality $b+1\le a$ are especially vulnerable. Derivation 3 uses the integer inequalities $2^{a+1}\ge n+1$, $2^{b+1}\ge a+1$ and retains the strict positive margin $2a$. The case $n=3$ realizes the small comparison $16>14$, so the margin is not manufactured by treating a strict inequality as equality. The argument applies throughout every interval of constant $a,b$ and through each jump.

Premises and composition. This consumes the examined binomial bound, the floor estimates and the two elementary monotone comparisons at the end of Derivation 3. The binomial convention beyond $n$ is retained. The asserted domain begins at $n=2$, exactly where the nested logarithm is defined.

Verdict: refutation-failed, with the same Sauer--Shelah premise. Both the specified integer $d$ and the two displayed upper estimates survive for every $n\ge2$.

Corollary 1.3(i)

Strongest attack and deduction. Rounding the real exponent down can lose nearly one, and the formula is negative for small $n$. Write $x=\ell-2\log_2(\ell+2)$, so $\omega=x-1$. If $\omega\le0$, then $n^\omega\le1\le F(n)$. Otherwise $x>1$ and $t=\lfloor x\rfloor$ satisfies $t\ge1$ and $\omega=x-1<t\le x\le\ell$. In particular,

$$t+2\log_2(t+2) \le x+2\log_2(\ell+2)=\ell.$$

Exponentiation gives $(t+2)^2 2^t\le2^\ell=\log_2 n$. The examined two-color threshold yields $F(n)\ge n^t\ge n^\omega$. This works when $x$ is an integer and when its fractional part is arbitrarily close to one; the supplied $-1$ is enough in both cases. At $n=2$, $\ell=0$ and $\omega=-3$, so the small case uses the first branch. Finally $\log_2(\ell+2)=o(\ell)$ as $\ell\to\infty$, giving $\omega\to\infty$.

Premises and composition. This consumes the examined two-color threshold and elementary logarithmic limits, with no Sauer--Shelah premise. To express the lower growth question on all positive integers, one can define the exponent at $n=1$ to be zero; $1^0=1\le F(1)$. The displayed formula itself is asserted only for $n\ge2$.

Verdict: refutation-failed. The explicit exponent, its complete stated domain and its divergent limit survive.

Corollary 1.3(ii)

Strongest attack and deduction. The natural choice $F(n)^{1/n}-1$ would give equality, not the claimed strict inequality. The source instead sets

$$\varepsilon(n)=F(n)^{1/n}-1+\frac1n.$$

Since $F(n)\ge1$ and $n\ge1$, $1+\varepsilon(n)=F(n)^{1/n}+1/n>F(n)^{1/n}>0$. Raising to the positive integer power $n$ gives the strict bound for every positive $n$, independent of any asymptotic estimate. In particular $F(1)=1$: color the two subsets differently for the upper bound, and use a singleton for the lower bound. Thus $\varepsilon(1)=1$ gives $1<2$.

For the limit, the examined explicit upper bound with $\lambda=\log_2 n$ and $D=\lambda+\log_2\lambda+3$ gives

$$0\le\log_2 F(n) \le\lambda^2+\lambda\log_2\lambda+2\lambda+\log_2 D =O(\lambda^2).$$

Since $\lambda^2/2^\lambda\to0$ and $n=2^\lambda$, it follows that $\log F(n)/n\to0$, whence $F(n)^{1/n}\to1$ and $\varepsilon(n)\to0$. Only the eventual estimate is used for the limit; it is not extended to undefined logarithms at $n=1$.

Premises and composition. This consumes the examined explicit upper bound, hence Sauer--Shelah through Derivation 3, as well as the finite extremum and positivity of $F$ and elementary exponential continuity. The function is allowed to be defined in terms of $F$; the assertion is existential, not an algorithm to evaluate $F$.

Verdict: refutation-failed, with the stated external premise for the limiting upper estimate. The strict inequality itself is verified at every positive integer.

Corollary 1.3(iii)

Strongest attack and deduction. Merely knowing $\log F=O((\log n)^2)$ would not establish the leading coefficient $1$ in this exponent comparison. It could only give an unspecified constant. The source uses the explicit bound instead. For $n\ge2$, logarithms in base $2$ and division by $\lambda=\log_2 n>0$ give

$$\begin{aligned} \omega(n) &\le D-1+\frac{\log_2 D}{\lambda}\\ &=\lambda+\log_2\lambda+2+\frac{\log_2 D}{\lambda}. \end{aligned}$$

After division of the excess by $\lambda$, the terms $\log_2\lambda/\lambda$, $2/\lambda$ and $\log_2 D/\lambda^2$ all tend to zero. Thus for each $\varepsilon>0$ their sum is at most $\varepsilon$ eventually, which gives the precise claimed coefficient. This is a comparison in the exponent's own units. It holds for any qualifying real function $\omega$, including one that is negative or irregular. The resulting threshold can even be chosen independently of that function; only $\varepsilon$ enters the error estimate.

Premises and composition. The premise $n^{\omega(n)}\le F(n)$ is the explicit hypothesis of this conditional assertion, not a conclusion assumed without proof. The other inputs are the examined explicit upper bound, hence Sauer--Shelah, and elementary logarithmic limits. The value at $n=1$ is irrelevant to the eventual conclusion.

Verdict: refutation-failed, with the stated external premise. The coefficient, quantifier over $\omega$, and quantifier over every $\varepsilon>0$ survive.

Corollary 1.3(iv)

Strongest attack and deduction. The thresholds from the lower theorem depend very strongly on both $k$ and $t$. An interchange of these quantifiers with $n\to\infty$ could be false. Indeed, allowing $k(n)\ge2^n$ permits injective colorings with $F_{k(n)}(n)=1$. That example refutes an unasserted varying-color version, not the fixed-color conclusion under examination.

Fix $k$. For any integer $t\ge1$, the polynomial theorem gives $F_k(n)\ge c_{k,t}n^{t+1}\ge n^t$ once $n\ge c_{k,t}^{-1}$. For each positive integer $n$, define

$$\omega_k(n)=\max\{t\in\{0,1,\ldots,n\}:n^t\le F_k(n)\}.$$

This is exactly the source's largest qualifying integer $t\le n$: $t=0$ qualifies, so no negative integer could alter that maximum. The set is nonempty and finite. For every fixed integer $t\ge1$, all integers $n\ge\lceil\max(t,c_{k,t}^{-1})\rceil$ have $\omega_k(n)\ge t$. Given any real target, choose a larger such integer $t$; this proves $\omega_k(n)\to\infty$ with the correct order of quantifiers. The lower inequality holds by the definition of the maximum at every positive $n$. At $n=1$ it has $\omega_k(1)=1$ and $1^{\omega_k(1)}=1\le F_k(1)=1$.

Set $\varepsilon_k=\varepsilon$ from (ii). Since unused colors are allowed, the already checked comparison $F_k(n)\le F(n)$ preserves the strict upper bound for every $n$, and $\varepsilon_k(n)\to0$. No change of coloring convention is used.

Premises and composition. The lower half consumes the examined fixed-$k$ polynomial bound and a finite maximum. The upper half consumes (ii) and the model comparison, hence Sauer--Shelah. The same upper function can serve every $k\ge2$; no analogous uniformity is claimed for the lower growth threshold.

Verdict: refutation-failed, with the stated external premise for the upper half. Both all-positive-$n$ inequalities and both limits survive for each fixed $k\ge2$.

Elementary limits and static examples

The logarithmic limits above require no experimental extrapolation. For example, on $\lambda\in[m,m+1]$, $\lambda^2/2^\lambda\le(m+1)^2/2^m$, and the latter sequence has successive ratio tending to $1/2$, so it decays geometrically after some point. Applying the same elementary comparison after writing a positive variable as $2^u$ gives $\log x/x\to0$. These limits imply all the stated $o(\ell)$ and normalized error estimates. The only additional limit operation is continuity of the exponential at zero.

The four supplied figures were checked as static mathematical text. The hole-cube labels add exactly the coordinate's generator along each indicated edge. In the block-chain example, the top at level $8$ has size $31$, the base for $A=\{2,4\}$ has size $26$, and the generators have sizes $2$ and $3$; the displayed member omits block $4$ and retains block $2$. The union example has omissions $\{2,4\}$ and $\{3,4\}$, whose common omission is $\{4\}$. The box example has corners $(1,1,2)$ and $(3,3,0)$, lower corner $(1,1,0)$, upper corner $(3,3,2)$, and shape $(r+2;2,2,2)$. These checks validate the examples' stated identities; no rendering or TeX execution is claimed or needed by the proofs.

Checklist verdicts

Every row below is a checklist verdict for each of the eight conclusion sections individually. The references specify the relevant argument for that conclusion; inapplicability is stated explicitly. The conclusion-specific sections above supply the associated strongest attack, composition and premises. This shared presentation does not leave any conclusion exempt from an item.

Canonical failure modes

Named patterns

Erdős-specific checklist

The neighboring statistical-correlation discipline is inapplicable because no statistical sample is used. The proof/evidence-coverage discipline is met by the complete internal coverage and the explicit external-premise boundary; the elementary examples have no claimed universal evidentiary force.

Premises, reading depth and verification order

No native L-claim or ledger row is consumed by the frozen statement or proof. This was checked against the actual mathematical files on 2026-10-15, including the consequence sentences. No live ledger was read, and no native standing or pending grade is used as a premise. Source result labels are not native claim identities.

Exact Sauer--Shelah interface

The external theorem used is:

For a family $\mathcal A$ of subsets of an $n$-element set, if its VC dimension is at most an integer $q\ge0$, then $|\mathcal A|\le\sum_{j=0}^q\binom nj$, with $\binom nj=0$ for $j>n$.

Here shattering means realizing every intersection trace on the test set. This is the finite cardinality form explicitly supplied in provenance_and_scope.md.txt and upper_proof.tex.txt, Lemma 5.1(c). The consuming specialization is $\mathcal A=\mathcal F$, a nonempty monochromatic union-closed family in the selected no-bad-tuple coloring, and $q=d-1\ge0$. Derivation 3 proves $\operatorname{vc}(\mathcal F)\le q$ from absence of an independent system. No closure hypothesis is required by Sauer--Shelah itself; union closure is used in the preceding step. For an empty family the cardinality bound is immediate without this interface.

The basis for reliance is the expressly commissioned classical Sauer--Shelah premise in this exact finite form. Actual reading depth is the supplied claim checked and its application verified. The original Sauer and Shelah papers are unread, and their proofs are not independently verified here. Their bibliographic citations, read from the supplied reference excerpt, are:

• N. Sauer, On the density of families of sets, Journal of Combinatorial Theory, Series A 13 (1972), no. 1, 145--147. DOI supplied by the source.
• S. Shelah, A combinatorial problem; stability and order for models and theories in infinitary languages, Pacific Journal of Mathematics 41 (1972), no. 1, 247--261. DOI supplied by the source.

No stronger variant, structural characterization, or assertion about intersection-closed families is taken from those references. If this external theorem is not granted, this report alone does not supply the binomial upper bound or the consequences that consume it. The lower polynomial bound, its threshold, Corollary (i), and the lower half of Corollary (iv) are independent of that external theorem.

Other inputs and credits

The internal box inequality is proved in the supplied text and independently rederived in Derivation 1. The source credits the method to W. T. Gowers, A new proof of Szemerédi's theorem, Geometric and Functional Analysis 11 (2001), no. 3, 465--588, DOI supplied by the source. That paper is unread here. Its attribution is preserved, but no theorem from it is an external premise.

The other foundational inputs are elementary finite counting and pigeonholing, finite product probability and the union bound, Cauchy--Schwarz and Jensen for nonnegative real quantities, and elementary logarithmic/exponential properties. Their exact roles are displayed: Cauchy--Schwarz and Jensen in the box and color-density bounds; counting and averaging in slices and gluing; product probability and the union bound in the bad-tuple construction; logarithms and limits in the threshold and corollary deductions. No claim of a new proof of these elementary principles is made.

The arXiv subject's supplied internal proofs have proof verified coverage for the assigned $F/F_k$ scope, relative to those elementary principles and the explicit Sauer--Shelah premise where used. This reading-depth statement covers the retained sections and deductions enumerated here, not the full paper, its original PDF, its other versions, or the excluded sublattice claims.

The analytic verification order is acyclic:

1. Cube definitions and permutation invariance establish the shape probability; the independently checked box inequality gives the grid estimate.
2. The grid estimate and permutation averaging give supersaturation. Supersaturation gives slices, and slices give a common arrangement by incidence counting.
3. The union rule, color pigeonholing and injective bases give gluing. It composes with that arrangement and the constant estimates to prove the polynomial bound, then the two-color threshold.
4. The threshold gives Corollary (i). The polynomial bound gives the lower half of Corollary (iv).
5. Private-element unions and the random-coloring argument give a coloring with no independent system inside any monochromatic union-closed family. Shattering and the external Sauer--Shelah interface give the binomial upper bound, then its explicit version.
6. The explicit upper version gives (ii) and (iii); (ii) and $F_k\le F$ give the upper half of (iv).

These are mathematical consumption relations within this examination, not promotions or assertions of native standing.

Reproduction, limitations and grading

The independent reasoning is retained in this report, especially the three derivations and the conclusion-specific attacks. All necessary finite examples and parameters are written explicitly. The exact fourteen supplied mathematical payloads and their neutral manifest are retained under subject/ beside the report. They are sufficient to reconstruct what was examined without a private archive or changing canonical page.

The reproduction route is analytic:

1. Read the retained model and statements, then the cube/gluing and box/supersaturation files against Derivation 1.
2. Reproduce the free-coordinate extension, incidence count, base injection and constant arithmetic in Derivation 2 against the retained lower proof.
3. Count the nonempty unions and bad events, verify the VC interface under the explicitly supplied Sauer--Shelah premise, and reproduce the strict floor estimates in Derivation 3 against the upper proof.
4. Check the eight separate attack/verdict sections, all four retained corollary deductions, and the explicit small cases and figure identities.

No mathematical computation was run or used. Accordingly there is no numerical checker, numerical output, executable rerun command, or computational failing-input obligation. The structurally disjoint computational-batch rule is inapplicable. This is a first independent examination, not a second-cycle countersign; no prior-cycle attacks were supplied. The alternative box induction and the explicit obstruction candidates are preserved mathematical work, not agreement with cached author results.

All assigned conclusions have the verdict refutation-failed, at the exact scopes stated individually above. No counterexample or unsupported essential internal deduction was found. The two auxiliary domain issues identified in the examination are fully localized: the probability application has a nonempty grid, and empty families are bounded directly. Neither requires a changed theorem, a changed constant, or an additional theorem premise. No correction to the frozen source was made or treated as part of the subject.

There is no unfinished internal proof obligation within the assigned $F/F_k$ scope. The limits of the verdict are the explicitly assumed Sauer--Shelah theorem, the exact frozen version and scope, and text-only inspection. Original Sauer/Shelah proofs, original-PDF fidelity, other source versions, the $f$ claims, formalization and public acceptance remain outside the examination. A change to the subject would require its own assessment.

These are the reviewer's mathematical verdicts. A distinct fresh grader must assess this unchanged report's contract and independence. This report gives no grade, native tier, source-acceptance determination, or problem-status assertion.