Effective bounds for the density Hales–Jewett problem for k ≥ 4

Establish effective upper bounds for the density Hales–Jewett problem for k ≥ 4 by showing that if A ⊆ {0,1,…,k−1}^n has no combinatorial lines of length k, then the density of A is at most a function with finitely many iterated logarithms in n, where the number of log applications is O_k(1).

Background

The density Hales–Jewett problem asks for the largest possible density of a subset of {0,1,…,k−1}^n avoiding combinatorial lines of length k. While vanishing density is known qualitatively via ergodic theory, quantitative bounds are challenging. The authors recently obtained ‘reasonable’ bounds (finite iterated logs) for k = 3 (Theorem 5.5).

Generalizing such effective bounds to k ≥ 4 would likely rely on advancing inverse theorem techniques for k-wise correlations, potentially building on Conjecture 3.13 and novel density-increment strategies.