On $\mathcal{F}$-multicolor Turán number of hypergraph graphs
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2025
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| _version_ | 1866917358609629184 |
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| author | Li, Ping |
| author_facet | Li, Ping |
| contents | The Ruzsa-Szemerédi $(6,3)$-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 Váli, 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 Turán 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 Váli. 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 $r=3,4$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2502_11869 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On $\mathcal{F}$-multicolor Turán number of hypergraph graphs Li, Ping Combinatorics The Ruzsa-Szemerédi $(6,3)$-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 Váli, 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 Turán 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 Váli. 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 $r=3,4$. |
| title | On $\mathcal{F}$-multicolor Turán number of hypergraph graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2502.11869 |