Rational exponents for hypergraph Turan problems

TL;DR

This paper demonstrates that for any rational number between 0 and k-1, there exists a finite hypergraph family such that the maximum number of edges in an n-vertex hypergraph avoiding this family scales as n to the power of k minus that rational number.

Contribution

The authors establish the existence of hypergraph families with Turán numbers scaling as a specific rational power of n, filling a gap in hypergraph extremal theory.

Findings

Existence of hypergraph families with Turán numbers of order n^{k-r} for rational r in [0, k-1]

Construction of hypergraph families matching these Turán bounds

Extension of Turán problem understanding to rational exponents

Abstract

Given a family of $k$-hypergraphs $\mathcal{F}$, $ex(n,\mathcal{F})$ is the maximum number of edges a $k$-hypergraph can have, knowing that said hypergraph has $n$ vertices but contains no copy of any hypergraph from $\mathcal{F}$ as a subgraph. We prove that for every rational $r$ between $0$ and $k-1$, there exists some finite family $\mathcal{F}$ of $k$-hypergraphs for which $ex(n,\mathcal{F})=\Theta(n^{k-r})$.