Sparse counting lemma for $K_4$

TL;DR

This paper proves the sparse counting lemma for the complete graph on four vertices ($K_4$), advancing the understanding of regularity methods in sparse random graphs.

Contribution

It establishes the $K_4$ case of the sparse counting lemma conjecture, which was previously known only for triangles.

Findings

Proves the sparse counting lemma for $K_4$ in sparse graphs.

Extends the sparse regularity method to a new case.

Provides tools for extremal results in random graph theory.

Abstract

The sparse analogue of Szemer\'edi's regularity method has played a central role in the development of extremal results for random graphs. While the sparse embedding lemma (the KLR conjecture) has been resolved, the corresponding sparse counting lemma remains widely open. The conjecture, formulated by Gerke, Marciniszyn, and Steger, states that for every fixed graph $H$ and any $\beta>0$, there exists $\varepsilon>0$ such that the following holds. Consider a balanced blow-up of $H$ with vertex classes of size $n$, where each pair corresponding to an edge of $H$ forms an $(\varepsilon)$-regular bipartite graph with exactly $m$ edges. Assume that $m$ is above the natural threshold $m \gg n^{2-1/m_2(H)}$, then all but a $\beta^m$ proportion of such graphs contain at least $(1-\delta)$ times the expected number of copies of $H$. At present, among the complete graphs, the conjecture is known only for $H=K_3$. In this paper, we establish the $H=K_4$ case of the conjecture.