Generalized Tur\'an problem with bounded matching number

TL;DR

This paper determines the maximum number of copies of a complete graph in large graphs that avoid certain subgraphs, extending previous results and providing exact values for specific cases.

Contribution

It extends the generalized Turán problem to include bounded matching numbers and determines exact maximum counts for copies of complete graphs under these constraints.

Findings

Exact value of ex(n, K_r, {K_{k+1}, M_{s+1}}) determined.

Extended previous results to general fixed graphs H.

Provided asymptotic bounds with an O(1) error term.

Abstract

For a graph $T$ and a set of graphs $\mathcal{H}$, let $\mbox{ex}(n,T,\mathcal{H})$ denote the maximum number of copies of $T$ in an $n$-vertex $\mathcal{H}$-free graph. Recently, Alon and Frankl~(arXiv2210.15076) determined the exact value of $\mbox{ex}(n,K_2,\{K_{k+1},M_{s+1}\})$, where $K_{k+1}$ and $M_{s+1}$ are complete graph on $k+1$ vertices and matching of size $s+1$, respectively. Soon after, Gerbner~(arXiv2211.03272) continued the study by extending $K_{k+1}$ to general fixed graph $H$. In this paper, we continue the study of the function $\mbox{ex}(n, T,\{H,M_{s+1}\})$ when $T=K_r$ for $r\ge 3$. We determine the exact value of $\mbox{ex}(n,K_r,\{K_{k+1},M_{s+1}\})$ and give the value of $\mbox{ex}(n,K_r,\{H,M_{s+1}\})$ for general $H$ with an error term $O(1)$.