On $\mathcal{F}$-multicolor Tur\'{a}n number of hypergraph 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 $r=3,4$.