[Paper Review] Triangle-free Subgraphs of Hypergraphs

Jiaxi Nie, Sam Spiro+1 more|arXiv (Cornell University)|Apr 23, 2020

Limits and Structures in Graph Theory25 references4 citations

TL;DR

This paper establishes a lower bound on the size of the largest triangle-free subhypergraph in $r$-uniform hypergraphs with maximum degree $ riangle$, showing it is at least $e(H)/ riangle^{(r-2)/(r-1)+o(1)}$. For $r=3$, this bound is tight up to the $o(1)$ term, and the result extends to random hypergraphs, where the extremal number of edges in a $T^3$-free subhypergraph is shown to be $ ilde{O}(p^{1/3}n^2)$ with high probability.

ABSTRACT

In this paper, we consider an analog of the well-studied extremal problem for triangle-free subgraphs of graphs for uniform hypergraphs. A loose triangle is a hypergraph $T$ consisting of three edges $e,f$ and $g$ such that $|e \cap f| = |f \cap g| = |g \cap e| = 1$ and $e \cap f \cap g = \emptyset$. We prove that if $H$ is an $n$-vertex $r$-uniform hypergraph with maximum degree $ riangle$, then as $ riangle ightarrow \infty$, the number of edges in a densest $T$-free subhypergraph of $H$ is at least \[ \frac{e(H)}{ riangle^{\frac{r-2}{r-1} + o(1)}}.\] For $r = 3$, this is tight up to the $o(1)$ term in the exponent. We also show that if $H$ is a random $n$-vertex triple system with edge-probability $p$ such that $pn^3 ightarrow\infty$ as $n ightarrow\infty$, then with high probability as $n ightarrow \infty$, the number of edges in a densest $T$-free subhypergraph is \[ \min\Bigl\{(1-o(1))p{n\choose3},p^{\frac{1}{3}}n^{2-o(1)}\Bigr\}.\] We use the method of containers together with probabilistic methods and a connection to the extremal problem for arithmetic progressions of length three due to Ruzsa and Szemerédi.

Motivation & Objective

• To generalize the classical Turán problem for triangle-free graphs to $r$-uniform hypergraphs, focusing on loose triangles.
• To determine the maximum number of edges in a $T^r$-free subhypergraph of a given $r$-graph with bounded maximum degree.
• To analyze the behavior of such extremal subgraphs in random $r$-uniform hypergraphs $G_{n,p}^r$.
• To establish tight bounds for the $r=3$ case and explore the gap in bounds for $r\geq4$.
• To connect the extremal problem to arithmetic progressions via the Ruzsa-Szemerédi framework.

Proposed method

• Use the method of hypergraph containers to bound the number of $T^r$-free subhypergraphs.
• Apply probabilistic methods to analyze the expected number of $T^r$-free subgraphs in random $r$-graphs $G_{n,p}^r$.
• Leverage the Ruzsa-Szemerédi connection between triangle-free linear hypergraphs and sets without 3-term arithmetic progressions.
• Derive upper bounds on the number of $T^r$-free $r$-graphs with $n$ vertices and $m$ edges using entropy-based counting and exponential bounds.
• Use Markov’s inequality to show that with high probability, no $T^r$-free subgraph exceeds a certain size in $G_{n,p}^r$.
• Establish asymptotic bounds on $\mathrm{ex}(G_{n,p}^r, T^r)$ by balancing the number of such subgraphs and their edge probabilities.

Experimental results

Research questions

• RQ1 What is the maximum number of edges in a $T^r$-free subhypergraph of an $r$-uniform hypergraph with maximum degree $\triangle$?
• RQ2 How does the extremal number of edges in a $T^3$-free subhypergraph behave in random $r$-graphs $G_{n,p}^r$?
• RQ3 Can the bound $\mathrm{ex}(G, T^r) \geq e(G)/\triangle^{(r-2)/(r-1)+o(1)}$ be improved or tightened for $r \geq 4$?
• RQ4 What is the correct exponent in the dependence on $\triangle$ for the extremal function $\mathrm{ex}(G, T^r)$ in the general case?
• RQ5 How does the extremal number of edges in $T^r$-free subgraphs of $G_{n,p}^r$ scale with $p$ for $r \geq 4$?

Key findings

• For any $r$-uniform hypergraph $G$ with maximum degree $\triangle$, the size of the largest $T^r$-free subhypergraph satisfies $\mathrm{ex}(G, T^r) \geq e(G)/\triangle^{(r-2)/(r-1)+o(1)}$ as $\triangle \to \infty$.
• For $r=3$, this bound is tight up to the $o(1)$ term in the exponent, as shown by a construction based on cliques $K_t^3$.
• In the random $3$-uniform hypergraph $G_{n,p}^3$, with high probability, $\mathrm{ex}(G_{n,p}^3, T^3) = \min\left\{(1-o(1))p\binom{n}{3}, p^{1/3}n^{2-o(1)}\right\}$.
• For $r \geq 4$, the best known lower bound is $\mathrm{ex}(G, T^r) = \Omega(\triangle^{-1/2}) \cdot e(G)$, but the true exponent remains open.
• The number of $T^r$-free $r$-graphs on $n$ vertices and $m$ edges is bounded by $\left(\frac{n^{r-1}}{m}\right)^{(1+o(1))m}$ for $m = n^{3-\delta}$, enabling extremal bounds.
• For $p = n^{-r+x}$ with $0 \leq x \leq r$, the expected extremal number of edges in $T^r$-free subgraphs of $G_{n,p}^r$ satisfies $f_r(x) = \lim_{n\to\infty} \log_n \mathbb{E}[\mathrm{ex}(G_{n,p}^r, T^r)]$, with $f_r(x) = x$ for $x \leq 3/2$ and $f_r(x) = x-1$ for $x > 4$, and bounds in between.