---

Source. Frankl–Rödl, published pp. 3–6. The results on this page are outside inputs to the 1990 paper; their proofs are not duplicated in this source folder.

Two-point super-Ramsey input. For every distance $a>0$, the two-point configuration at distance $a$ has exponential finite density witnesses as in definitions. Frankl–Rödl p. 3, Corollary 2.3, attributes this to P. Frankl and R. M. Wilson, Intersection theorems with geometric consequences, Combinatorica 1 (1981), 357–368, and states $c=2$, $\epsilon=0.2$. The proof here needs only the existence of positive constants; it does not independently derive or use those particular numerical values. This input is stronger than a chromatic-number bound alone.

Joint-partition input. Fix integers $r\ge2$, $q\ge2$ and a real $\eta>0$. Let $l_0,\ldots,l_{q-1}$ be positive integers summing to $n$. Let $M=(m_{j_1\ldots j_r})$, indexed by $0\le j_i<q$, be a nonnegative integer array with every entry at least $\eta n$, and with each one-coordinate marginal equal to the same vector $(l_0,\ldots,l_{q-1})$:

$$\sum_{\mathbf j:j_i=a}m_{\mathbf j}=l_a \quad(1\le i\le r,\ 0\le a<q).$$

There exists $0<\epsilon<1$, depending only on the fixed parameters, such that every family $\mathcal K$ of ordered partitions of $[n]$ into cells of these sizes with

$$|\mathcal K|\ge(1-\epsilon)^n\frac{n!}{\prod_a l_a!}$$

contains partitions $(A_0^{(i)},\ldots,A_{q-1}^{(i)})$, $1\le i\le r$, with

$$\left|\bigcap_{i=1}^r A_{j_i}^{(i)}\right|=m_{j_1\ldots j_r} \quad\text{for every }(j_1,\ldots,j_r).$$

This is the equal-family existence consequence of Frankl–Rödl, Forbidden intersections, Transactions of the AMS 300 (1987), 259–286, Theorem 1.16, printed p. 265 (author-hosted published PDF). The original theorem is stronger: it allows separate families and marginals, and gives a positive lower bound for the number of prescribed patterns. Choose its $\gamma$ in $(0,1)$ and a smaller entry threshold, for example $\eta/2$, to satisfy its strict entry inequality. The marginal conditions make the full-family pattern count positive: allocate disjoint coordinate blocks of the specified sizes. Its lower bound therefore guarantees existence. The same equal-family statement is explicitly restated as Theorem 2.2 on p. 219 of Frankl–Rödl, Strong Ramsey properties of simplices, Israel Journal of Mathematics 139 (2004), 215–236 (published PDF). Both statements were checked visually against those versions. The full original proof chain of the 1987 theorem is now compiled at the canonical source linked above. It remains external to the 1990 paper.

Two-point hyper-Ramsey input. For every $a>0$ and $\delta>0$, the configuration of two points at distance $a$ has super-Ramsey witnesses on $S(a/2+\delta,n)$ for every sufficiently large $n$. Frankl–Rödl p. 6 attributes this to Frankl–Wilson and cites V. Rödl, On a problem in combinatorial geometry, Discrete Mathematics 45 (1983), 129–131 (DOI), for an explicit statement. Frankl–Rödl 2004 p. 221 repeats the input. The original 1983 proof has not been audited here; the full radius-sensitive input stays external. It is used for the closing brick result, not the main super-Ramsey simplex proof.

Inputs expanded locally. The finite negative-type criterion and the positive-uniformity case of the modular intersection theorem quoted on pp. 3 and 5 are proved on negative_type_criterion and modular_independence. These are supplied elementary expansions of the quoted inputs, not claimed reproductions of their original papers.

Bears on. #174.