Bounds for graph regularity and removal lemmas

We show, for any positive integer k, that there exists a graph in which any equitable partition of its vertices into k parts has at least ck^2/\log^* k pairs of parts which are not \epsilon-regular, where c,\epsilon>0 are absolute constants. This bound is tight up to the constant c and addresses a question of Gowers on the number of irregular pairs in Szemer'edi’s regularity lemma. In order to gain some control over irregular pairs, another regularity lemma, known as the strong regularity lemma, was developed by Alon, Fischer, Krivelevich, and Szegedy. For this lemma, we prove a lower bound of wowzer-type, which is one level higher in the Ackermann hierarchy than the tower function, on the number of parts in the strong regularity lemma, essentially matching the upper bound. On the other hand, for the induced graph removal lemma, the standard application of the strong regularity lemma, we find a different proof which yields a tower-type bound. We also discuss bounds on several related regularity lemmas, including the weak regularity lemma of Frieze and Kannan and the recently established regular approximation theorem. In particular, we show that a weak partition with approximation parameter \epsilon may require as many as 2^{\Omega(\epsilon^{-2})} parts. This is tight up to the implied constant and solves a problem studied by Lov'asz and Szegedy.

💡 Research Summary

The paper investigates the quantitative limits of several fundamental graph regularity results, providing matching lower bounds that demonstrate the optimality of known upper bounds. The authors first address a long‑standing question concerning the dependence of Szemerédi’s regularity lemma on the parameter η, which measures the fraction of irregular pairs allowed in a partition. They construct, for every positive integer k, a graph such that any equitable partition of its vertex set into k parts contains at least c·k²/ log* k pairs of parts that are not (ε, δ)‑regular, where c, ε, δ are absolute constants and log* denotes the iterated logarithm. This shows that the number of parts required when η is small grows as a tower of twos of height proportional to η⁻¹, matching the tower‑type upper bound proved by Gowers and others. The construction uses a hierarchy of refinements P₁,…,P_s, each refined exponentially, and inserts independent random graphs G_i on the parts of P_i with carefully chosen edge probabilities. Random bipartitions of adjacent parts are then used to create edges that enforce a large number of irregular pairs with high probability.

Next, the paper turns to the strong regularity lemma of Alon, Fischer, Krivelevich, and Szegedy, which simultaneously guarantees an equitable partition A and an equitable refinement B that is both ε‑close to A and f(|A|)‑regular. The known proof yields a wowzer‑type bound (one level above towers) on the number of parts in B. The authors prove a complementary lower bound of the same wowzer magnitude, establishing that for any decreasing function f and any small ε, there exists a graph G such that any pair of equitable partitions A, B satisfying q(B) ≤ q(A)+ε and with B f(|A|)‑regular must have |A|, |B| at least W, where W is defined recursively by a wowzer‑type sequence (W₁ = 1, W_{ℓ+1}=T(2^{-70ε⁵}/f(W_ℓ)), with T the tower function). Consequently, the wowzer upper bound is essentially optimal.

The authors then revisit the induced graph removal lemma, which states that a graph with few induced copies of a fixed H can be made induced‑H‑free by adding or deleting at most εn² edges. The standard proof again invokes the strong regularity lemma, inheriting a wowzer‑type bound on the dependence of δ on ε. By exploiting the lower bound for the strong lemma and a refined analysis, the authors present an alternative proof that avoids the strong lemma altogether. Their argument combines a weak regularity lemma with a regular approximation theorem, yielding a tower‑type bound for δ⁻¹ as a function of ε⁻¹—significantly improving the quantitative dependence compared with the previous wowzer‑type result.

Finally, the paper resolves a problem posed by Lovász and Szegedy concerning the weak regularity lemma of Frieze and Kannan. While the weak lemma guarantees that an ε‑approximation can be achieved with O(ε⁻²) parts, the authors prove that this bound is tight up to constants: there exist graphs for which any weak partition achieving ε‑approximation must use at least 2^{Ω(ε⁻²)} parts. This lower bound matches the known upper bound, confirming the optimality of the weak regularity lemma’s quantitative statement.

Overall, the work delivers a comprehensive picture of the quantitative landscape of graph regularity lemmas. By constructing explicit extremal graphs and employing probabilistic methods, the authors demonstrate that the tower‑type dependence on ε for the ordinary regularity lemma, the wowzer‑type dependence for the strong regularity lemma, and the exponential dependence for the weak regularity lemma are all unavoidable. Moreover, the new tower‑type proof of the induced removal lemma shows that, despite the strong lemma’s inherent wowzer complexity, certain applications can be achieved with substantially better bounds. The results settle several open questions and firmly establish the optimality of the known bounds across a suite of central combinatorial tools.