Sparse counting lemma for $K_4$

Abstract: The sparse analogue of Szemerédi'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 $β>0$, 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 $β^m$ proportion of such graphs contain at least $(1-δ)$ 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.

• The paper establishes the first complete sparse counting lemma for K4, rigorously confirming the Gerke-Marciniszyn-Steger conjecture.
• It introduces novel auxiliary graph constructions and a multi-layer regularity method to accurately count K4 copies in sparse graphs.
• The result provides asymptotically sharp lower bounds on K4 counts, guiding future work on sparse graph embeddings and larger cliques.

Sparse Counting Lemma for $K_4$: Authoritative Summary

Context and Motivation

The sparse regularity method, originating in the extension of Szemerédi's regularity lemma to sparse graphs, underpins much of modern extremal combinatorics, especially concerning random graphs. While dense graph analogues — such as embedding and counting lemmas — are well-established, transferring their power to sparse settings has required new definitions and technical tools. Embedding lemmas in the sparse regime (e.g., the KŁR Conjecture) have seen complete resolution via the hypergraph container method [Balogh-Morris-Samotij; Saxton-Thomason]. However, the corresponding sparse counting lemma has remained largely elusive, particularly cases beyond $K_3$.

The counting lemma conjectured by Gerke, Marciniszyn, and Steger posits that for any fixed $H$ and edge density above the natural threshold $n^{-1/m_2(H)}$, most sufficiently regular sparse graphs contain nearly the expected number of copies of $H$. The only previously resolved complete graph case is $H=K_3$.

This paper establishes, for the first time, the full form of the sparse counting lemma for $H=K_4$, closing a significant gap in the transfer of regularity techniques from the dense to the sparse setting for cliques of size $4$.

Main Results

The Formal Statement

Let $K_4$ denote the complete graph on 4 vertices. The primary theorem verifies the Gerke-Marciniszyn-Steger sparse counting conjecture for $H=K_4$:

For every $K_3$0, there exist constants $K_3$1 and $K_3$2 such that for all $K_3$3, all but a $K_3$4 fraction of $K_3$5-regular blown-up copies with edge densities at least $K_3$6 contain at least $K_3$7 times the expected number of canonical $K_3$8 copies.

This matches, up to constant factors, what would be expected in the binomial random graph with corresponding edge densities.

Technical Advances

The arguments hinge on a careful generalization of several tools:

• Extension of regularity inheritance results and lower-regularity for graphs with unbalanced vertex class sizes.
• Construction of auxiliary bipartite graphs ("path-Aux graphs") to transfer lower-regularity properties among various bipartite slices of the blow-up.
• A refined inductive argument, leveraging the aforementioned auxiliary constructions and precise control on the distribution of canonical subgraphs.

Key technical devices include:

• A multi-level extension of the regularity/probabilistic method, with error probabilities compounded across successive layers.
• Explicit combinatorial and probabilistic lemmas ensuring the smallness of exceptional families at each step, bounding the fraction of graphs failing the conclusion.

Implications

Theoretical Impact

The resolution of the $K_3$9 case of the sparse counting lemma is a nontrivial strengthening over previously known methods:

• It achieves asymptotically sharp lower bounds on the number of $H$0 copies for almost all regular subgraphs at density $H$1, matching the natural random threshold derived from $H$2.
• Unlike prior partial results that either demanded higher minimum densities, or only bounded the fraction of missing copies by a constant rather than an arbitrarily small $H$3, this result realizes all parameters in their essentially optimal ranges.

This progress cements the program of extending regularity-based extremal and anti-Ramsey principles into the sparse regime for cliques up to size $H$4.

Algorithmic and Practical Ramifications

While focused on extremal questions and not algorithmics directly, the result underpins the validity of using regularity-based methods to count and embed $H$5 subgraphs in large, sparse, pseudorandom structures.

Potential applications include:

• Random graph models: Sharp concentration of $H$6 counts in random-like sparse graphs.
• Property testing: Bounds on instance-wise reliability for property-testing algorithms based on counting arguments in sparse graphs.
• Sparse pseudorandom graph design: Guidance for network construction where local $H$7 density is critical, and resilience to adversarial perturbations is desired.

Limitations and Obstacles for $H$8 and Beyond

The methods, though powerful for $H$9, face an explicit combinatorial barrier for $n^{-1/m_2(H)}$0 and larger cliques. The auxiliary graph constructions grow increasingly complex; the number of feasible triangle (or higher simplex) packings explodes far faster than the number of ways to choose underlying edge sets, especially at the $n^{-1/m_2(H)}$1 threshold $n^{-1/m_2(H)}$2. As elucidated in the conclusion, this prevents the direct iteration of the approach for $n^{-1/m_2(H)}$3, since the space of exceptional structures swamps the space of realizable edge-supporting configurations.

Thus, while the $n^{-1/m_2(H)}$4 case appears to be the critical threshold for this methodology, significant new ideas seem necessary for larger complete graphs.

Conclusion

This paper provides a comprehensive resolution of the sparse counting lemma for $n^{-1/m_2(H)}$5, closing a longstanding conjecture in sparse extremal combinatorics. The result establishes that highly regular, nearly threshold-density sparse graphs almost always contain nearly the predicted number of $n^{-1/m_2(H)}$6 copies. These findings reinforce the regularity method as a robust tool for both dense and sparse regimes (at least for $n^{-1/m_2(H)}$7), and demarcate clear boundaries for the difficulties posed by larger complete subgraphs.

Future directions will likely require novel analytic or probabilistic ideas to extend sparse counting results to $n^{-1/m_2(H)}$8 and beyond. The demonstrated methodology herein is, however, likely to inspire both further progress and applications, both combinatorial and algorithmic, across discrete mathematics and random graph theory.

Reference: "Sparse counting lemma for $n^{-1/m_2(H)}$9" (2603.29938)