Triple systems with no three triples spanning at most five points

TL;DR

This paper determines the maximum size of certain 3-uniform hypergraphs avoiding small subconfigurations, confirming a conjecture for the case k=5 and establishing the limit as 1/5.

Contribution

It proves the conjecture for k=5, showing the limit exists and equals 1/5, extending previous results for k=4.

Findings

Maximum triples on n points without three spanning at most five points is approximately n^2/5.

Confirmed the Brown-Erdős-Sós conjecture for k=5.

Used optimization and approximate H-decompositions in the proof.

Abstract

We show that the maximum number of triples on $n$~points, if no three triples span at most five points, is $(1\pm o(1))n^2/5$. More generally, let $f^{(r)}(n;k,s)$ be the maximum number of edges of an $r$-uniform hypergraph on $n$~vertices not containing a subgraph with $k$~vertices and $s$~edges. In 1973, Brown, Erd\H{o}s and S\'os conjectured that the limit $\lim_{n\to \infty}n^{-2}f^{(3)}(n;k,k-2)$ exists for all~$k$. They proved this for $k=4$, where the limit is $1/6$ and the extremal examples are Steiner triple systems. We prove the conjecture for $k=5$ and show that the limit is~$1/5$. The upper bound is established via a simple optimisation problem. For the lower bound, we use approximate $H$-decompositions of~$K_n$ for a suitably defined graph~$H$.