On Tur\'an problems with bounded matching number

TL;DR

This paper investigates the maximum number of edges in large graphs that avoid a certain subgraph and have a limited matching number, providing near-complete solutions and several exact results.

Contribution

It determines the maximum edge count for $F$-free graphs with bounded matching number, advancing understanding of Turán-type problems with new exact and near-exact results.

Findings

Exact maximum edge counts for specific cases

Near-complete solutions for general fixed $F$ and $s$

Several new exact results in the area

Abstract

Very recently, Alon and Frankl initiated the study of the maximum number of edges in $n$-vertex $F$-free graphs with matching number at most $s$. For fixed $F$ and $s$, we determine this number apart from a constant additive term. We also obtain several exact results.