The Erd\H{o}s-Hajnal hypergraph Ramsey problem

TL;DR

This paper resolves a long-standing conjecture in hypergraph Ramsey theory by precisely determining the tower growth rate of the function g_k(t,n), confirming the conjectured exponential tower behavior in almost all cases.

Contribution

It establishes the exact tower growth rate of g_k(t,n) for nearly all parameter ranges, settling the Erd ext{"o}s-Hajnal conjecture with precise bounds.

Findings

Confirmed the tower growth rate for g_k(t,n) matches the conjecture in most cases.

Determined the exact polynomial power within the tower for half of the cases.

Provided new bounds that improve understanding of hypergraph Ramsey numbers.

Abstract

Given integers $2\le t \le k+1 \le n$, let $g_k(t,n)$ be the minimum $N$ such that every red/blue coloring of the $k$-subsets of $\{1, \ldots, N\}$ yields either a $(k+1)$-set containing $t$ red $k$-subsets, or an $n$-set with all of its $k$-subsets blue. Erd\H{o}s and Hajnal proved in 1972 that for fixed $2\le t \le k$, there are positive constants $c_1$ and $c_2$ such that $$ 2^{c_1 n} < g_k(t, n) < twr_{t-1} (n^{c_2}),$$ where $twr_{t-1}$ is a tower of 2's of height $t-2$. They conjectured that the tower growth rate in the upper bound is correct. Despite decades of work on closely related and special cases of this problem by many researchers, there have been no improvements of the lower bound for $2<t<k$. Here we settle the Erd\H{o}s-Hajnal conjecture in almost all cases in a strong form, by determining the correct tower growth rate, and in half of the cases we also determine the correct power of $n$ within the tower. Specifically, we prove that if $2<t<k-1$ and $k - t$ is even, then $$g_k(t, n) = twr_{t-1} (n^{k-t+1 + o(1)}).$$ Similar results are proved for $k - t$ odd.