Unavoidable order-size pairs in hypergraphs -- positive forcing density

TL;DR

This paper investigates the existence of specific order-size pairs in hypergraphs that are unavoidable, establishing the pair (6,10) as a positive forcing density example for 3-uniform hypergraphs and conjecturing its uniqueness.

Contribution

The paper identifies the pair (6,10) as a positive forcing density in 3-uniform hypergraphs and proposes that it may be the only such pair, advancing understanding of hypergraph induced substructure constraints.

Findings

(6,10) has positive forcing density for r=3.

Necessary conditions for positive forcing density are established.

Conjecture that (6,10) is the unique such pair for r=3.

Abstract

Erd\H{o}s, F\"uredi, Rothschild and S\'os initiated a study of classes of graphs that forbid every induced subgraph on a given number $m$ of vertices and number $f$ of edges. Extending their notation to $r$-graphs, we write $(n,e) \to_r (m,f)$ if every $r$-graph $G$ on $n$ vertices with $e$ edges has an induced subgraph on $m$ vertices and $f$ edges. The \emph{forcing density} of a pair $(m,f)$ is $$ \sigma_r(m,f) =\left. \limsup\limits_{n \to \infty} \frac{|\{e : (n,e) \to_r (m,f)\}|}{\binom{n}{r}} \right. .$$ In the graph setting it is known that there are infinitely many pairs $(m, f)$ with positive forcing density. Weber asked if there is a pair of positive forcing density for $r\geq 3$ apart from the trivial ones $(m, 0)$ and $(m, \binom{m}{r})$. Answering her question, we show that $(6,10)$ is such a pair for $r=3$ and conjecture that it is the unique such pair. Further, we find necessary conditions for a pair to have positive forcing density, supporting this conjecture.