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.