Erd\H os--Hajnal problem for $H$-free hypergraphs

TL;DR

This paper investigates the minimum edge count in H-free graphs with high chromatic number, linking bounds from Ramsey and Turán theories to derive new bounds on this parameter.

Contribution

It establishes a connection between Ramsey and Turán bounds and the minimum edges in H-free graphs with large chromatic number, providing new bounds.

Findings

Bounds on $m_H(r)$ derived from Ramsey numbers

Bounds on $m_H(r)$ derived from Turán numbers

Implications for hypergraph extremal problems

Abstract

This paper deals with the minimum number $m_H(r)$ of edges in an $H$-free graph with the chromatic number more than $r$. We show how bounds on Ramsey and Tur\'an numbers imply bounds on $m_H(r)$.