Sparse Counting Lemma (Gerke–Marciniszyn–Steger) Conjecture

Establish the sparse counting lemma for arbitrary fixed graphs H as formulated by Gerke, Marciniszyn, and Steger: For every fixed graph H and any β > 0 and δ > 0, determine constants ε > 0, C > 0, and n0 > 0 such that for all n ≥ n0 and m ≥ C n^{2-1/m2(H)}, the number of ε-regular H-blow-ups with vertex classes of size n and exactly m edges per bipartite pair that contain fewer than (1 − δ) n^{|V(H)|} (m/n^2)^{|E(H)|} canonical copies of H is at most β^m times {n^2 choose m}^{|E(H)|}. Equivalently, prove that |(H, n, m, δ) ∩ (H, n, m, ε)| ≤ β^m {n^2 choose m}^{|E(H)|} under these conditions.

Background

The paper studies the sparse analogue of the counting lemma within Szemerédi’s regularity method. While the KŁR embedding conjecture has been resolved via hypergraph containers, its counting counterpart—asserting near-expected counts of copies of a fixed graph H in ε-regular blow-ups at the natural threshold density—has not been settled in general.

The conjecture seeks optimal parameters simultaneously: (i) edge density m/n^2 above the threshold n^{-1/m2(H)}, (ii) a failure probability at most β^m for arbitrarily small β, and (iii) an (1 − δ) factor of the expected number of copies of H. Prior work established various partial results, including the full statement for H = K3. This paper proves the case H = K4, leaving the conjecture open in general and particularly for larger complete graphs.