📝 SummarXiv

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 originating from three distinct triangle-copies. Gowers and Janzer extended the result to uniform hypergraphs by showing that there exist at least $n^{k-o(1)}$ edge-disjoint copies of $K_r^{(k)}$ with no $K_r^{(k)}$ formed by edges from distinct $K_r^{(k)}$-copies. Recently, Imolay, Karl, Nazy and V\'{a}li explored a broader question: for two simple graphs $F$ and $G$, determine the maximum number of edge-disjoint copies of $F$ in a set of $n$ vertices such that there is no copy of $G$ whose edges come from different $F$-copies. This maximum number, denoted by $ex_F(n,G)$, is termed the {\em $F$-multicolor Tur\'{a}n number} of $G$. In this paper, we first show 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 $\mathcal{F}$, which extends a key result by Imolay et al. We also establish the corresponding supersaturation and blow-up phenomena. In the non-degenerate regime, we show that $\frac{n^k}{v(F)^k}+o(n^{k-1})\leq ex_{\mathcal{F}}(n,\mathcal{G})\leq e(\mathcal{F})^{-1}ex(n,\mathcal{G})+o(n^k)$. In particular, $ex_{\mathcal{F}}(n,\mathcal{G})=e(\mathcal{F})^{-1}ex(n,\mathcal{G})+o(n^k)$ holds if $\mathcal{F}$ is a complete $k$-part $k$-uniform hypergraph with parts of equal size. For degenerate case, we show that $ex_{\mathcal{F}}(n,\mathcal{G})=n^{k-o(1)}$ whenever $\mathcal{G}$ contains $H_3^{(k)}$ as a subgraph, where $H_3^{(k)}$ denotes a $k$-uniform hypergraph on $k+1$ vertices with three edges.