On $\mathcal{F}$-multicolor Tur\'{a}n number of hypergraph graphs

TL;DR

This paper investigates the maximum number of edge-disjoint hypergraph copies avoiding certain substructures, extending classical Turán problems to hypergraphs with new bounds, characterizations, and stability results.

Contribution

It generalizes the multicolor Turán number to hypergraphs, providing necessary and sufficient conditions, bounds, and characterizations for various special cases and structures.

Findings

Established when $ex_{\mathcal{F}}(n,\mathcal{G})=o(n^k)$ based on homomorphisms.

Derived bounds and stability conditions for hypergraph Turán numbers.

Characterized $\\\\\\\\\mathcal{F}$ for non-attainment of upper bounds in specific intersecting graphs.

Abstract

The Ruzsa-Szemer\'{e}di $(6,3)$-problem can be equivalently stated as determining the maximum number of edge-disjoint triangles on $n$ vertices such that no triangle is formed by edges from three distinct triangle-copies. Gowers and Janzer extended this problem by establishing an analogous result for complete graphs. A natural generalization of the two results, first introduced by Imolay, Karl, Nagy and V\'{a}li, asks for the maximum number of edge-disjoint copies of a graph $F$ on $n$ vertices such that no copy of $G$ is formed by edges originating from distinct $F$-copies. This maximum number, denoted by $ex_F(n,G)$, is called the {\em $F$-multicolor Tur\'{a}n number} of $G$. This paper focuses on the setting of uniform hypergraphs. We first prove that for $k$-uniform hypergraphs $\mathcal{G}$ and $\mathcal{F}$, $ex_{\mathcal{F}}(n,\mathcal{G})=o(n^k)$ if and only if there exists a homomorphism from $\mathcal{G}$ to $\mathcal{F}$. For degenerate case, we show that $ex_{\mathcal{F}}(n,\mathcal{G})=n^{k-o(1)}$ whenever $\mathcal{G}$ contains a $k$-uniform tight triangle. These results extend previous results. We further establish corresponding supersaturation and blowup statements. In the non-degenerate setting, we derive matching lower and upper bounds for $ex_{\mathcal{F}}(n,\mathcal{G})$. We give a necessary and sufficient condition for $ex_{\mathcal{F}}(n,\mathcal{G})$ to fail to attain the upper bound, under the assumption that the extremal graphs for $\mathcal{G}$ are stable. As an application, we refine a result due to Imolay, Karl, Nagy and V\'{a}li. Furthermore, we completely characterize $\mathcal{F}$ for which $ex_{\mathcal{F}}(n,\mathcal{G})$ does not attain the upper bound when $\mathcal{G}$ is one of the three special intersecting graphs: Fano plane, extended triangle and $r$-book of $r$-edges with $r=3,4$.