Forbidding matching as trace in uniform hypergraphs

TL;DR

This paper investigates the maximum size of uniform hypergraphs avoiding certain trace sub-hypergraphs, providing asymptotic bounds and exact values for specific cases involving matchings and cliques.

Contribution

It establishes sharp asymptotic bounds and exact values for Turán numbers related to traces of matchings and cliques in hypergraphs, extending previous work to trace-based problems.

Findings

Sharp asymptotic upper bounds for x}_r(n, Tr_r(M_{s+1}))

Exact values of x}_3(n, Tr_3(M_{s+1}))

Bounds for generalized Ture1n numbers for cliques in trace-avoiding hypergraphs

Abstract

We say a hypergraph $\mathcal{H}$ contains a hypergraph $\mathcal{G}$ as trace if there exists a vertex subset $S \subseteq V(\mathcal{H})$ such that $|S| = |V(\mathcal{G})|$ and $\{e \cap S: e \in E(\mathcal{H})\}$ contains $\mathcal{G}$ as a sub-hypergraph. We use $\mathrm{ex}_r(n, \mathrm{Tr}_r(\mathcal{G}))$ to denote the maximum number of hyperedges in an $r$-uniform hypergraph on $n$ vertices not containing $\mathcal{G}$ as a trace. The study of Tur\'{a}n numbers for traces was initiated by Mubayi and Zhao who studied the case when $\mathcal{G}$ is a complete graph. Let $M_{s+1}$ denote the graph of a matching with $s+1$ edges. In this paper, we give the upper bound of $\mathrm{ex}_r(n, \mathrm{Tr}_r(M_{s+1}))$ which is sharp asymptotically. When $r=3$, we give the exact value of $\mathrm{ex}_3 (n, \mathrm{Tr}_3 (M_{s+1}))$. We also consider the generalized Tur\'{a}n number in the case of matching. That is, the maximum number of copies of clique $\mathcal{K}_t^r$ in hypergraphs forbidding $\mathrm{Tr}_r (M_{s+1})$ as a trace. We give an upper bound which is sharp asymptotically and when $r=3$, we give the exact value. The Tur\'{a}n number of forbidding a matching and the other graph is another well studied topic initiated by Alon and Frankl. We also consider an analogue problem for the trace version, i.e., forbidding trace of matching and trace of complete graph as subgraphs.