On the Maximum Number of Edges in Hypergraphs with Fixed Matching and Clique Number

TL;DR

This paper determines the maximum number of edges in k-uniform hypergraphs with given matching and clique constraints, providing exact bounds for large n and specific parameters.

Contribution

It establishes precise maximum edge counts for hypergraphs with fixed matching and clique numbers, including special cases and bounds for large n.

Findings

Maximum edges for hypergraphs with fixed matching and clique numbers.

Exact solutions for special cases q=(s+1)k-2 and k=2.

Bounds for n in relation to k, s, and q.

Abstract

For a $k$-graph $\mathcal{F}\subset \binom{[n]}{k}$, the clique number of $\mathcal{F}$ is defined to be the maximum size of a subset $Q$ of $[n]$ with $\binom{Q}{k}\subset \mathcal{F}$. In the present paper, we determine the maximum number of edges in a $k$-graph on $[n]$ with matching number at most $s$ and clique number at least $q$ for $n\geq 8k^2s$ and for $q \geq (s+1)k-l$, $n\leq (s+1)k+s/(3k)-l$. Two special cases that $q=(s+1)k-2$ and $k=2$ are solved completely.