An efficient asymmetric removal lemma and its limitations

TL;DR

This paper advances the asymmetric removal framework by providing a regularity-free proof that poly(ε) n^5 copies of C5 must appear in graphs ε-far from triangle-free, and by introducing K3-abundance to classify when a fixed graph H must appear polynomially many times. It shows that all odd cycles C_{2ℓ+1} (ℓ≥2) are K3-abundant, but, conditional on a Ruzsa genus conjecture, not all triangle-free tripartite graphs are K3-abundant, via a Ruzsa–Szemerédi-type construction. The authors develop a synthesis of combinatorial, number-theoretic, probabilistic, and Ramsey-type techniques, including a robust notion of strongly genus-one graphs and a pseudorandom-graph framework, to exhibit triangle-free tripartite graphs that are not K3-abundant. They also connect these structural results to property testing by bounding sampling complexity for monotone graph properties and discuss the implications for larger cliques and higher chromatic numbers. Overall, the work separates the abundance landscape for odd cycles from that of general graphs and opens questions about efficient asymmetric removal lemmas for higher chromatic numbers and explicit constructions.

Abstract

The triangle removal states that if $G$ contains $\varepsilon n^2$ edge-disjoint triangles, then $G$ contains $δ(\varepsilon)n^3$ triangles. Unfortunately, there are no sensible bounds on the order of growth of $δ(\varepsilon)$, and at any rate, it is known that $δ(\varepsilon)$ is not polynomial in $\varepsilon$. Csaba recently obtained an asymmetric variant of the triangle removal, stating that if $G$ contains $\varepsilon n^2$ edge-disjoint triangles, then $G$ contains $2^{-\mathrm{poly}(1/\varepsilon)}\cdot n^5$ copies of $C_5$. To this end, he devised a new variant of Szemerédi's regularity lemma. We obtain the following results: - We first give a regularity-free proof of Csaba's theorem, which improves the number of copies of $C_5$ to the optimal number $\mathrm{poly}(\varepsilon)\cdot n^5$. - We say that $H$ is $K_3$-abundant if every graph containing $\varepsilon n^2$ edge-disjoint triangles has $\mathrm{poly}(\varepsilon)\cdot n^{|V(H)|}$ copies of $H$. It is easy to see that a $K_3$-abundant graph must be triangle-free and tripartite. Given our first result, it is natural to ask if all triangle-free tripartite graphs are $K_3$-abundant. Our second result is that assuming a well-known conjecture of Ruzsa in additive number theory, the answer to this question is negative. Our proofs use a mix of combinatorial, number-theoretic, probabilistic, and Ramsey-type arguments.

Theorems & Definitions (31)

• Theorem 1.1
• Theorem 1.2: Csaba Csaba
• Theorem 1.3
• Definition 1.4: $K_3$-abundant
• Conjecture 1.5: Ruzsa Ruzsa
• Theorem 1.6
• Theorem 1.7: GGR
• Theorem 1.8
• Lemma 2.1
• Corollary 2.2