Improved bounds on the extremal function of hypergraphs

TL;DR

This paper establishes a connection between graph and matrix extremal functions, introduces a new method to bound hypergraph extremal functions, and improves existing bounds for specific hypergraph classes.

Contribution

It develops a novel approach linking hypergraph extremal functions to multidimensional matrices, leading to tighter bounds for permutation hypergraphs.

Findings

Proved an equivalence between graph and matrix extremal functions.

Developed a new method to bound hypergraph extremal functions.

Improved the bound for d-permutation hypergraphs from O(n^{d-1}) to 2^{O(k)}n^{d-1}.

Abstract

A fundamental problem in pattern avoidance is describing the asymptotic behavior of the extremal function and its generalizations. We prove an equivalence between the asymptotics of the graph extremal function for a class of bipartite graphs and the asymptotics of the matrix extremal function. We use the equivalence to prove several new bounds on the extremal functions of graphs. We develop a new method to bound the extremal function of hypergraphs in terms of the extremal function of their associated multidimensional matrices, improving the bound of the extremal function of $d$-permutation hypergraphs of length $k$ from $O(n^{d-1})$ to $2^{O(k)}n^{d-1}$.