On $\mathcal{F}$-multicolor Turán number of hypergraph graphs

TL;DR

This work develops a comprehensive framework for the $\mathcal{F}$-multicolor Turán problem in $k$-uniform hypergraphs, establishing a sharp dichotomy: $ex_{\mathcal{F}}(n,\mathcal{G})=\Theta(n^k)$ if no homomorphism from $\mathcal{G}$ to $\mathcal{F}$ exists, and $ex_{\mathcal{F}}(n,\mathcal{G})=o(n^k)$ otherwise. In the degenerate case, when $\mathcal{G}$ contains a $k$-uniform tight triangle, the growth becomes $n^{k-o(1)}$, with supersaturation and blowup phenomena analyzed via the Hypergraph Regularity Lemma and Counting Lemma. The non-degenerate regime yields matching upper and lower bounds, and the authors provide necessary and sufficient conditions for when the natural upper bound is not attained under stability. Applications include refining results of Imolay, Karl, Nazy, and Váli and complete classifications for when $ex_{\mathcal{F}}(n,\mathcal{G})$ fails to attain the bound for specific intersecting graphs such as the Fano plane, extended triangle, and $r$-books.

Abstract

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, Nazy 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, Nazy 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 $r=3,4$.

Figures (1)

• Figure 1: The left graph is known as the Fano plane (the circle and segments represent edges), denoted by $\mathbb{F}$. The middle graph $\mathcal{C}_3^{(2k)}$ is called the extended $2r$-uniform triangle, which is a $2k$-uniform hypergraph with $V(\mathcal{C}_3^{(2k)})=[3k]$ and $E(\mathcal{C}_3^{(2k)})=\{[2k],[k+1,3k]_{\mathbb{N}},[3k]-[k+1,2k]_{\mathbb{N}}\}$. The right graph $\mathcal{B}_k^{(k)}$ is called the $k$-book of $k$ pages, which is a $k$ uniform hypergraph with $V(\mathcal{B}_k^{(k)})=[k-1]\cup\{v_i:i\in[k]\}$ and $E(\mathcal{B}_k^{(k)})=\{[k-1]\cup\{v_i\}:i\in[k]\}\cup\{v_i:i\in[k]\}$.

Theorems & Definitions (46)

• Theorem 1.1: Alon and Shapira, Alon-Shapira
• Remark 1.1
• Conjecture 1: Alon and Shapira, Alon-Shapira
• Theorem 1.2: Gowers and Janzer, Gowers
• Corollary 1.3
• Theorem 1.4: Imolay, Karl, Nazy and Váli, IKNV
• Theorem 1.5: Kovács and Nagy, C5-C3-1
• Theorem 1.6
• Proposition 1
• Theorem 1.7