On forbidding graphs as traces of hypergraphs

TL;DR

This paper investigates the maximum size of 3-uniform hypergraphs that do not contain a specific graph as a trace, improving bounds for certain graphs and providing exact results for large hypergraphs.

Contribution

It improves existing bounds for hypergraphs avoiding certain graph traces and determines exact bounds for large hypergraphs when avoiding book graphs.

Findings

Improved bound for hypergraphs avoiding $C_4$ as a trace.

Exact bounds established for large hypergraphs avoiding book graphs.

Extended understanding of trace-avoiding hypergraph structures.

Abstract

We say that a hypergraph $\mathcal{H}$ contains a graph $H$ as a trace if there exists some set $S\subset V(\mathcal{H})$ such that $\mathcal{H}|_S=\{h\cap S: h\in E(\mathcal{H})\}$ contains a subhypergraph isomorphic to $H$. We study the largest number of hyperedges in 3-uniform hypergraphs avoiding some graph $F$ as trace. In particular, we improve a bound given by Luo and Spiro in the case $F=C_4$, and obtain exact bounds for large $n$ when $F$ is a book graph.