Graph removal lemmas

The graph removal lemma states that any graph on n vertices with o(n^{v(H)}) copies of a fixed graph H may be made H-free by removing o(n^2) edges. Despite its innocent appearance, this lemma and its extensions have several important consequences in number theory, discrete geometry, graph theory and computer science. In this survey we discuss these lemmas, focusing in particular on recent improvements to their quantitative aspects.

💡 Research Summary

The survey paper provides a comprehensive overview of the graph removal lemma, its proofs, quantitative refinements, and a wide range of applications across mathematics and computer science. The lemma states that for any fixed graph H, a large host graph G on n vertices that contains only o(n^{v(H)}) copies of H can be made H‑free by deleting at most o(n^2) edges. The authors begin by recalling the classical proof based on Szemerédi’s regularity lemma. By partitioning G into a bounded number of regular pairs, one can show that if the number of H‑copies is sublinear in the natural scaling, then a small fraction of edges inside irregular or low‑density pairs suffices to eliminate every copy of H.

The early quantitative bounds derived from this approach are extremely weak: the relationship between the allowed copy density δ and the edge‑removal budget ε is of tower‑type, i.e., δ ≈ tower(O(1/ε)). Such bounds are far from practical for algorithmic applications. The paper then surveys a series of breakthroughs that dramatically improve this dependence. Fox (2011) introduced an “energy‑increment” method combined with dependent random choice, reducing the bound to δ ≥ ε^{O(log 1/ε)}. Subsequent work by Fox and Zhao (2015) refined the analysis further, achieving δ ≥ ε^{c log 1/ε} for a universal constant c. These results replace the monstrous tower function with a quasi‑polynomial relationship, making the lemma usable in property‑testing algorithms and approximation schemes.

The authors also discuss the hypergraph removal lemma, the higher‑dimensional analogue. While the regularity machinery becomes far more intricate, recent advances by Gowers–Wolf and Conlon‑Fox‑Zhao have introduced weak regularity concepts and refined counting lemmas that yield bounds of the form δ ≥ ε^{c (log 1/ε)^{k‑1}} for k‑uniform hypergraphs. Although still super‑polynomial, these bounds represent a substantial step toward practical hypergraph applications.

A substantial portion of the survey is devoted to applications. In additive combinatorics, the graph removal lemma provides short proofs of Roth’s theorem on three‑term arithmetic progressions and, via the hypergraph version, of Szemerédi’s theorem on long arithmetic progressions. In discrete geometry, it underlies results about incidences and uniform point‑line configurations. In theoretical computer science, the lemma is a cornerstone of property testing: Alon and Shapira (2005) showed that any hereditary graph property is testable with a constant‑time algorithm that queries O(1/ε) edges, a direct consequence of the removal lemma’s “local‑to‑global” principle. The paper also highlights recent attempts to bypass the regularity lemma entirely, aiming for removal lemmas with linear or near‑linear dependencies between δ and ε.

The survey concludes by outlining open problems: (1) achieving optimal δ–ε relations without invoking regularity; (2) tightening hypergraph bounds to approach polynomial dependence; (3) designing explicit, efficient edge‑removal procedures that match the existential guarantees. These challenges sit at the intersection of extremal combinatorics, additive number theory, and algorithm design, promising rich avenues for future research.