A note on the Tur\'an number for the traces of hypergraphs

TL;DR

This paper investigates the maximum size of hypergraphs avoiding certain trace subgraphs, providing improved bounds for specific cases involving stars and small complete bipartite graphs.

Contribution

It improves lower bounds for hypergraphs avoiding traces of stars and refines upper bounds for 3-uniform hypergraphs avoiding small complete bipartite graphs.

Findings

Enhanced lower bounds for hypergraphs avoiding star traces.

Improved upper bounds for 3-uniform hypergraphs avoiding small $K_{2,t}$ traces.

Identification of some cases where bounds are optimal.

Abstract

Let $\mathcal{H}$ be an $r$-uniform hypergraph and $F$ be a graph. We say $\mathcal{H}$ contains $F$ as a trace if there exists some set $S\subseteq V(\mathcal{H})$ such that $\mathcal{H}|_{S}:=\{E\cap S: E\in E(\mathcal{H})\}$ contains a subgraph isomorphic to $F.$ Let $ex_r(n,Tr(F))$ denote the maximum number of edges of an $n$-vertex $r$-uniform hypergraph $\mathcal{H}$ which does not contain $F$ as a trace. In this paper, we improve the lower bounds of $ex_r(n,Tr(F))$ when $F$ is a star, and give some optimal cases. We also improve the upper bound for the case when $\mathcal{H}$ is $3$-uniform and $F$ is $K_{2,t}$ when $t$ is small.