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Source. Proposition 1, p. 1, with its proof on pp. 1–2, in §1 "A signed laminar estimate" (pp. 1–2) of A Two-Copy Proof of Erdős Problem 126 (2026), a three-page preliminary exposition with no printed author, posted at https://www.erdosproblems.com/static/126-proof.pdf; the edition read is identified on the source card.

Statement

Setting (p. 1). $V$ has $n\geq2$ elements and $\mathcal P$ has $r$ elements. For each $p\in\mathcal P$, $\mathcal F_p$ is a finite labelled family of subsets of $V$; every member has at least two elements, any two supports are disjoint or one contains the other, and each label $B$ has a weight $w_p(B)\geq0$. Each vertex has a sign $\sigma_p(i)\in\{-1,1\}$ for each $p$, and

$$M_p(i,j)=\sum_{\substack{B\in\mathcal F_p\\i,j\in B}}w_p(B),\qquad C(i,j)=\sum_{\sigma_p(i)\ne\sigma_p(j)}M_p(i,j),\qquad R(i,j)=\sum_{\sigma_p(i)=\sigma_p(j)}M_p(i,j),$$

the last two sums running over $p\in\mathcal P$. A symmetric kernel $C$ is conditionally negative semidefinite if $\sum_{i,j}x_ix_jC(i,j)\leq0$ whenever $\sum_ix_i=0$.

Proposition 1 (p. 1, quoted). "If $C$ is conditionally negative semidefinite and $R(i,j)<C(i,j)$ $(i\neq j)$, then $n\ll r^2$."

Explicit constant (derived on this page, not printed). Tracking the constants in the printed proof gives $n\leq3r^2$. The same constant appears in signed_family_card_bound in the pinned formal module named on the source card.

Read depth. Claims checked: the setting, the statement and the proof on pp. 1–2 were read clause by clause, and the constant $3$ was derived here from the proof's two estimates. Nothing here is independently reviewed.

Proof sketch

Pp. 1–2. With $\Sigma=\sum_{i,j}C(i,j)$ and $T=\sum_p\sum_iM_p(i,i)$, the proof shows $\Sigma\ll rT$ and $nT\ll r\Sigma$, displayed as (1). Each $M_p$ is positive semidefinite, so $Q=R-C$, the sum of the sign-twisted $M_p$, is positive semidefinite with negative off-diagonal entries and trace $T$; testing $Q$ on the sign vectors of each $p$ bounds the total mass of every $M_p$ by $2T$, and with $\sum_{i,j}R(i,j)\geq\Sigma$ this gives $\Sigma\leq rT$. For the second estimate, a [[arithmetic_functions/adamczewski_2026_erdos126/two_copy_matching|two-copy matching]] sends each vertex to another member of its smallest support, using each target at most twice, and conditional negativity tested on $ne_i-\mathbf1$ and $n(e_i+e_j)-2\mathbf1$ gives pointwise bounds on $C(i,j)$, displayed as (6); together they give $n^2\sum_iM_p(i,i)\leq3n\Sigma$ for each $p$, so $nT\leq3r\Sigma$. Since $\Sigma>0$, combining the two estimates gives $n\leq3r^2$.

Dependencies

The [[arithmetic_functions/adamczewski_2026_erdos126/two_copy_matching|two-copy matching]], displayed as (3) and (4) in the proof, which rests on Hall's marriage theorem.

Bears on

• Problem 126: the proposition is the abstract estimate that the main theorem applies to prime-power residue families to bound the size of a set by the number of primes dividing its pair sums. On its own it says nothing about integers.