On forbidding graphs as traces of hypergraphs
Dániel Gerbner, Michael E. Picollelli
TL;DR
The paper investigates extremal questions for graphs as traces in $3$-uniform hypergraphs, focusing on the forbidden-trace problem $\mathrm{ex}_3(n,\mathrm{Tr}(F))$. The authors connect this to generalized Turán problems and supersaturation, and develop a shadow-graph framework to translate trace constraints into graph-based counting problems. They prove a sharp upper bound $\mathrm{ex}_3(n,\mathrm{Tr}(C_4)) \le \frac{1+\sqrt{2}}{4} n^{3/2} + o(n^{3/2})$ and establish exact asymptotics $\mathrm{ex}_3(n,\mathrm{Tr}(B_t))=\left\lfloor \dfrac{(n-1)^2}{4}\right\rfloor$ for large $n$, with the extremal construction adding a common vertex to every edge of a maximum bipartite graph on $n-1$ vertices. The methods hinge on shadow-graph analysis, a decomposition into light and heavy edges, and a Luo–Spiro-type lemma, supplemented by removal and stability arguments for the book graphs. These results advance understanding of graph-based hypergraphs and provide tight bounds that tie trace forbiddenness to classical Turán-type extremal behavior.
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.