Erdős–Hajnal Conjecture on off-diagonal hypergraph Ramsey numbers

Prove that for every fixed pair of integers 4 ≤ k < s there exists a constant c > 0 such that the off-diagonal hypergraph Ramsey number satisfies r_k(s,n) ≥ twr_{k−1}(c n), where the tower function is defined by twr_1(x)=x and twr_{i+1}(x)=2^{twr_i(x)}.

Background

The conjecture asserts that, for fixed uniformity k ≥ 4 and fixed clique size s > k, the lower bounds for r_k(s,n) should achieve the full tower height k−1 with linear argument, matching the known upper-bound tower height. It refines classical stepping-up lower bounds and is verified only in certain ranges of s (e.g., s ≥ ⌈5k/2⌉−3 or s ≥ k+3).

This paper improves lower bounds in specific small-gap cases (notably r_4(6,n) and, via stepping-up, r_k(k+2,n)) but the general conjecture remains open. Establishing the conjecture would settle the tower-height behavior for all fixed 4 ≤ k < s.

References

A fundamental and important conjecture about $r_k(s,n)$ was proposed by Erd\H{o}s and Hajnal . For fixed integers $4\le k<s$, it holds that $r_k(s,n)\ge \operatorname{twr}_{k-1}(\Omega(n)).$

— A Note on Generalized Erdős-Rogers Problems  (2604.02835 - Du et al., 3 Apr 2026) in Conjecture (Erdős–Hajnal), Section 1 (Introduction)

Applying this lemma yields the following corollary, which establishes Conjecture~\ref{E-H-C} in the case $s=k+2$ for $k\ge 4$, leaving only the case $r_4(5,n)\ge 2{2{cn}$ open.

— The Erdős--Hajnal hypergraph Ramsey problem for $r_4(6,n)$  (2609.26563 - Du et al., 22 Sep 2026) in Introduction, immediately following Corollary 1.3