[Paper Review] Hypergraphs with few Berge paths of fixed length between vertices

Zhiyang He, Michael Tait|arXiv (Cornell University)|Jul 26, 2018

Limits and Structures in Graph Theory24 references3 citations

TL;DR

This paper extends the classical even-cycle problem in extremal graph theory to hypergraphs by studying the maximum number of hyperedges in an $r$-uniform hypergraph where no pair of vertices has more than $t$ Berge paths of fixed length $k$. Using probabilistic and algebraic geometry methods, the authors establish that the extremal number of edges is $\Theta(n^{1+1/k})$ when $t$ is sufficiently large, generalizing known results for graphs to hypergraphs and confirming the order of magnitude for Berge-$\Theta_{k,t}$-free hypergraphs.

ABSTRACT

In this paper we study the maximum number of hyperedges which may be in an $r$-uniform hypergraph under the restriction that no pair of vertices has more than $t$ Berge paths of length $k$ between them. When $r=t=2$, this is the even-cycle problem asking for $\mathrm{ex}(n, C_{2k})$. We extend results of Füredi and Simonovits and of Conlon, who studied the problem when $r=2$. In particular, we show that for fixed $k$ and $r$, there is a constant $t$ such that the maximum number of edges can be determined in order of magnitude.

Motivation & Objective

• To generalize the Turán-type problem for even cycles ($C_{2k}$) from graphs to $r$-uniform hypergraphs by forbidding multiple Berge paths of fixed length between vertex pairs.
• To determine the extremal number of hyperedges in $r$-uniform hypergraphs where no pair of vertices has more than $t$ Berge paths of length $k$.
• To establish the order of magnitude of the extremal function $\mathrm{ex}_r(n, \Theta_{k,t}^B)$ for fixed $k$ and $r$, and to show that this order is tight when $t$ is sufficiently large.

Proposed method

• Adopt a probabilistic construction using random polynomials over finite fields to generate hypergraphs with controlled path counts between vertex pairs.
• Use algebraic geometry tools, including varieties and polynomial systems, to bound the number of degenerate and non-simple walks that could mimic valid Berge paths.
• Apply the Schwartz-Zippel lemma and Markov’s inequality to show that with high probability, the constructed hypergraph contains $\Omega_{k,r}(n^{1+1/k})$ edges and avoids $\Theta_{k,t}^B$ subhypergraphs for large $t$.
• Use a reduction lemma inspired by Győri and Lemons to relate $r$-uniform hypergraphs to $2$-uniform graphs, enabling the transfer of known graph-theoretic bounds to hypergraphs.
• Define $S_\sigma$ as the set of valid Berge paths of length $k$ with a fixed sequence of hyperedge indices, and use complexity bounds on associated algebraic varieties to control path counts.
• Apply Chebyshev’s inequality to show concentration of the number of edges around its expected value, ensuring the existence of a hypergraph with the desired extremal properties.

Experimental results

Research questions

• RQ1 What is the maximum number of hyperedges in an $r$-uniform hypergraph where no pair of vertices has more than $t$ Berge paths of length $k$?
• RQ2 Does the extremal function $\mathrm{ex}_r(n, \Theta_{k,t}^B)$ grow as $\Theta(n^{1+1/k})$ for fixed $k$ and $r$ when $t$ is sufficiently large?
• RQ3 How does the extremal number of edges depend on the uniformity $r$ and the number of allowed paths $t$?
• RQ4 Can the order of magnitude of the extremal function be determined for hypergraphs that forbid multiple Berge paths between vertex pairs, generalizing the even-cycle problem?
• RQ5 What is the dependence of the extremal function on $t$ when $t$ is large, and how does this compare to the graph case?

Key findings

• For fixed $k$ and $r$, there exists a constant $c_{r,k,t}$ such that $\mathrm{ex}_r(n, \Theta_{k,t}^B) \leq c_{r,k,t} n^{1+1/k}$, generalizing the Füredi–Simonovits bound to hypergraphs.
• When $t$ is large enough relative to $k$ and $r$, the extremal function satisfies $\mathrm{ex}_r(n, \Theta_{k,t}^B) = \Omega_{k,r}(n^{1+1/k})$, showing the order of magnitude is tight.
• The construction uses random polynomials over $\mathbb{F}_q$ to generate $r$-uniform hypergraphs on $n = rN$ vertices with $\Omega(N^{1+1/k})$ edges and no $\Theta_{k,C_{k,r}+1}^B$ subhypergraphs.
• The number of Berge paths of length $k$ between any pair of vertices is bounded by $C_{k,r}$ in the random construction, and this bound is used to show that with high probability, no pair exceeds this path count.
• The method relies on algebraic geometry to control degenerate walks and apply the Schwartz-Zippel lemma, ensuring that the number of valid paths remains bounded.
• The variance of the number of edges is shown to be $O(\mathbb{E}[X])$, so by Chebyshev’s inequality, the number of edges concentrates around its expectation, guaranteeing the existence of a hypergraph with $\Omega(n^{1+1/k})$ edges and the desired forbidden substructure property.