[Paper Review] Random Tur\'an theorem for hypergraph cycles

Dhruv Mubayi, Liana Yepremyan|arXiv (Cornell University)|Jul 20, 2020

Limits and Structures in Graph Theory22 references4 citations

TL;DR

This paper establishes the typical Turán number for linear even cycles $C_{2 heta}^{(r)}$ in the random $r$-uniform hypergraph $G_{n,p}^{(r)}$, determining the threshold behavior up to polylogarithmic factors for all but a shrinking range of $p$. It introduces a novel codegree-dependent supersaturation result that strengthens prior bounds and enables precise estimates across a wide range of edge probabilities.

ABSTRACT

Given $r$-uniform hypergraphs $G$ and $H$ the Tur\\'an number $\ m ex(G, H)$ is the maximum number of edges in an $H$-free subgraph of $G$. We study the typical value of $\ m ex(G, H)$ when $G=G_{n,p}^{(r)}$, the Erd\\H{o}s-R\\'enyi random $r$-uniform hypergraph, and $H=C_{2\\ell}^{(r)}$, the $r$-uniform linear cycle of length $2\\ell$. The case of graphs ($r=2$) is a longstanding open problem that has been investigated by many researchers. We determine $\ m ex(G_{n,p}^{(r)}, C_{2\\ell}^{(r)})$ up to polylogarithmic factors for all but a small interval of values of $p=p(n)$ whose length decreases as $\\ell$ grows. Our main technical contribution is a balanced supersaturation result for linear even cycles which improves upon previous such results by Ferber-Mckinley-Samotij and Balogh-Narayanan-Skokan. The novelty is that the supersaturation result depends on the codegree of some pairs of vertices in the underlying hypergraph. This approach could be used to prove similar results for other hypergraphs $H$.

Motivation & Objective

• To determine the typical value of the Turán number $\mathrm{ex}(G_{n,p}^{(r)}, C_{2 heta}^{(r)})$ in the Erdős–Rényi random $r$-uniform hypergraph $G_{n,p}^{(r)}$.
• To extend the understanding of random Turán problems beyond bipartite graphs, focusing on $r$-uniform linear even cycles.
• To develop a stronger supersaturation theorem that accounts for vertex codegrees, improving upon existing results for hypergraph cycles.
• To establish asymptotically almost sure bounds on the size of the largest $C_{2 heta}^{(r)}$-free subgraph in $G_{n,p}^{(r)}$ across a wide range of $p$.
• To provide a framework applicable to other $r$-partite $r$-graphs beyond even cycles.

Proposed method

• Introduce a balanced supersaturation result for linear even cycles that depends on the codegree of vertex pairs in the hypergraph.
• Use the codegree-dependent supersaturation to derive tight bounds on $\mathrm{ex}(G_{n,p}^{(r)}, C_{2 heta}^{(r)})$ for $r$-graphs.
• Apply the container method and probabilistic inequalities (e.g., Chernoff bounds, Markov’s inequality) to control the number of small cycles in random hypergraphs.
• Construct explicit $C_{2 heta}^{(r)}$-free subgraphs via Steiner systems and blowups of high-girth 3-graphs to establish lower bounds.
• Leverage Bertrand’s postulate to construct partial Steiner systems with desired properties for the construction of sparse, high-girth subgraphs.
• Use concentration inequalities to show that the number of isolated edges in random blowups of high-girth 3-graphs is concentrated around its expectation, enabling lower bound constructions.

Experimental results

Research questions

• RQ1 What is the typical value of $\mathrm{ex}(G_{n,p}^{(r)}, C_{2 heta}^{(r)})$ for $r$-uniform random hypergraphs $G_{n,p}^{(r)}$?
• RQ2 How does the codegree of vertex pairs affect the supersaturation of linear even cycles in hypergraphs?
• RQ3 Can a stronger supersaturation result be developed that accounts for codegree to improve Turán-type bounds in random hypergraphs?
• RQ4 What is the threshold behavior of $\mathrm{ex}(G_{n,p}^{(r)}, C_{2 heta}^{(r)})$ across different ranges of $p=n^{-\alpha}$?
• RQ5 To what extent can the codegree-dependent supersaturation method be generalized to other $r$-partite $r$-graphs?

Key findings

• The paper determines $\mathrm{ex}(G_{n,p}^{(r)}, C_{2 heta}^{(r)})$ up to polylogarithmic factors for all but a shrinking interval of $p$ as $\ell$ increases.
• A new codegree-dependent supersaturation result is proven, improving upon prior work by Ferber-McKinley-Samotij and Balogh-Narayanan-Skokan.
• For $p \ll n^{-1+1/(2\ell-1)}$, the Turán number satisfies $\mathrm{ex}(G_{n,p}^{(r)}, C_{2 heta}^{(r)}) = (1-o(1))e(G_{n,p}^{(r)})$ a.a.s.
• For $p \gg n^{-1+1/(2\ell-1)}$, the Turán number is bounded by $O(p^{1/\ell}n^{1+1/\ell})$ up to polylogarithmic factors.
• A construction using blowups of high-girth 3-graphs yields a $C_4^{(3)}$-free subgraph with $\Omega(p^{1/4}n^{3/2})$ edges a.a.s., valid for $p \gg n^{-2}$.
• The lower bound $\Omega(n^{4/3})$ is achieved at $p = n^{-2/3}$, matching the threshold for the 3-uniform case.