# Polynomial to exponential transition in Ramsey theory

## TL;DR

This paper proves a longstanding conjecture in hypergraph Ramsey theory, determining the exact value of a function related to hypergraph independence numbers and edge counts, and constructs quasirandom hypergraphs with optimal density properties.

## Contribution

It settles Erdős and Hajnal's conjecture for all s ≥ k ≥ 4 and constructs optimal quasirandom hypergraphs, advancing understanding of hypergraph extremal functions.

## Key findings

- Confirmed the recursive formula for h^{(k)}(s) for all s ≥ k ≥ 4.
- Constructed quasirandom hypergraphs with positive density and sharp upper density bounds.
- Resolved a question by Bhat and Röd regarding hypergraph density properties.

## Abstract

Given $s \ge k\ge 3$, let $h^{(k)}(s)$ be the minimum $t$ such that there exist arbitrarily large $k$-uniform hypergraphs $H$ whose independence number is at most polylogarithmic in the number of vertices and in which every $s$ vertices span at most $t$ edges. Erd\H os and Hajnal conjectured (1972) that $h^{(k)}(s)$ can be calculated precisely using a recursive formula and Erd\H os offered \$500 for a proof of this. For $k=3$ this has been settled for many values of $s$ including powers of three but it was not known for any $k\geq 4$ and $s\geq k+2$. Here we settle the conjecture for all $s \ge k \ge 4$. We also answer a question of Bhat and R\"odl by constructing, for each $k \ge 4$, a quasirandom sequence of $k$-uniform hypergraphs with positive density and upper density at most $k!/(k^k-k)$. This result is sharp.