Supersaturation of Even Linear Cycles in Linear Hypergraphs

TL;DR

This paper extends classical supersaturation results from graphs to linear hypergraphs, establishing lower bounds on the number of even linear cycles in hypergraphs with many edges, and introduces a reduction technique for supersaturation problems.

Contribution

It generalizes Simonovits' supersaturation theorem to r-uniform linear hypergraphs and develops a reduction lemma for almost-regular host graphs.

Findings

Establishes lower bounds for even linear cycles in hypergraphs with many edges.

Provides a self-contained proof including the case r=2.

Introduces a reduction lemma useful for other supersaturation problems.

Abstract

A classic result of Erd\H{o}s and, independently, of Bondy and Simonovits says that the maximum number of edges in an $n$-vertex graph not containing $C_{2k}$, the cycle of length $2k$, is $O( n^{1+1/k})$. Simonovits established a corresponding supersaturation result for $C_{2k}$'s, showing that there exist positive constants $C,c$ depending only on $k$ such that every $n$-vertex graph $G$ with $e(G)\geq Cn^{1+1/k}$ contains at least $c\left(\frac{e(G)}{v(G)}\right)^{2k}$ many copies of $C_{2k}$, this number of copies tightly achieved by the random graph (up to a multiplicative constant). In this paper, we extend Simonovits' result to a supersaturation result of $r$-uniform linear cycles of even length in $r$-uniform linear hypergraphs. Our proof is self-contained and includes the $r=2$ case. As an auxiliary tool, we develop a reduction lemma from general host graphs to almost-regular host graphs that can be used for other supersaturation problems, and may therefore be of independent interest.