Counting substructures II: triple systems

TL;DR

This paper establishes tight lower bounds on the number of specific substructures within triple systems, extending hypergraph Turán problem results and utilizing advanced combinatorial tools like the hypergraph removal lemma.

Contribution

It provides the first such bounds for hypergraphs, generalizing earlier theorems and solving conjectures related to the minimum number of substructures in dense triple systems.

Findings

Proved tight lower bounds for substructure counts in triple systems.

Extended classical results to hypergraphs, including the Fano plane.

Used hypergraph removal lemma and stability results in proofs.

Abstract

For various triple systems $F$, we give tight lower bounds on the number of copies of $F$ in a triple system with a prescribed number of vertices and edges. These are the first such results for hypergraphs, and extend earlier theorems of Bollob\'as, Frankl, F\"uredi, Keevash, Pikhurko, Simonovits, and Sudakov who proved that there is one copy of $F$. A sample result is the following: F\"uredi-Simonovits and independently Keevash-Sudakov settled an old conjecture of S\'os by proving that the maximum number of triples in an $n$ vertex triple system (for $n$ sufficiently large and even) that contains no copy of the Fano plane is $p(n)={n/2 \choose 2}n.$ We prove that there is an absolute constant $c$ such that if $n$ is sufficiently large and $1 \le q \le cn^2$, then every $n$ vertex triple system with $p(n)+q$ edges contains at least $6q({n/2 \choose 4}+(n/2 -3){n/2 \choose 3}$$ copies of the Fano plane. This is sharp for $q\le n/2-2$. Our proofs use the recently proved hypergraph removal lemma and stability results for the corresponding Tur\'an problem.