A new randomized algorithm for the Erdos--Hajnal problem

TL;DR

This paper introduces a new randomized algorithm that improves bounds on the minimum number of edges in hypergraphs with high chromatic number, generalizing previous results and simplifying the proof for the case r=2.

Contribution

A novel randomized algorithm providing improved bounds for the Erdős–Hajnal problem and its generalization, with a simpler proof for the case r=2.

Findings

New bound for m(n,r) as c n^{(r-1)/r} r^{n-1}

Improves previous bounds across a wide parameter range

Simplifies the proof for the case r=2

Abstract

In 1961 Erd\H{o}s and Hajnal introduced the quantity $m(n)$ as the minimum number of edges in an $n$-uniform hypergraph with chromatic number at least 3. The best known lower and upper bounds for $ m(n) $ are $ c_1 \sqrt{\frac{n}{\ln n}} 2^n$ and $c_2 n^2 2^n$ respectively. The lower bound is due to Radhakrishnan and Srinivasan (see \cite{RS}). A natural generalization for $ m(n) $ is the quantity $ m(n,r) $, which is the minimum number of edges in an $n$-uniform hypergraph with chromatic number at least $r+1$. In this work, we present a new randomized algorithm yielding a bound $ m(n,r) \ge c n^{\frac{r-1}{r}} r^{n-1} $, which improves upon all the previous bounds in a wide range of the parameters $ n, r $. Moreover, for $ r = 2 $, we get exactly the same bound as in the work \cite{RS} of Radhakrishnan and Srinivasan, and our proof is simpler.