On the van der Waerden numbersw(2;3,t)

Abstract

We present results and conjectures on the van der Waerden numbers w(2;3,t) and on the new palindromic van der Waerden numbers pdw(2;3,t). We have computed the new number w(2;3,19) = 349, and we provide lower bounds for 20 <= t <= 39, where for t <= 30 we conjecture these lower bounds to be exact. The lower bounds for 24 <= t <= 30 refute the conjecture that w(2;3,t) <= t^2, and we present an improved conjecture. We also investigate regularities in the good partitions (certificates) to better understand the lower bounds. Motivated by such reglarities, we introduce *palindromic van der Waerden numbers* pdw(k; t_0,...,t_{k-1}), defined as ordinary van der Waerden numbers w(k; t_0,...,t_{k-1}), however only allowing palindromic solutions (good partitions), defined as reading the same from both ends. Different from the situation for ordinary van der Waerden numbers, these "numbers" need actually to be pairs of numbers. We compute pdw(2;3,t) for 3 <= t <= 27, and we provide lower bounds, which we conjecture to be exact, for t <= 35. All computations are based on SAT solving, and we discuss the various relations between SAT solving and Ramsey theory. Especially we introduce a novel (open-source) SAT solver, the tawSolver, which performs best on the SAT instances studied here, and which is actually the original DLL-solver, but with an efficient implementation and a modern heuristic typical for look-ahead solvers (applying the theory developed in the SAT handbook article of the second author).Comment: Second version 25 pages, updates of numerical data, improved formulations, and extended discussions on SAT. Third version 42 pages, with SAT solver data (especially for new SAT solver) and improved representation. Fourth version 47 pages, with updates and added explanation

New lower bounds for van der Waerden numbers

“…, n D+1 and θ. A representative case (but not the only one we need to consider) would be ) , with the coefficients of q having size ∼ L −2 so that q is bounded in size by O (1). Note that N −4/D ≈ L −4r .…”

“…Thus we have a problem of roughly the following type: given a quadratic form q : Z D+1 → R with coefficients of size ∼ L −2 , show that on the box [L] D+1 it takes at least one value on some given interval of length L −4r . Without further information, this is unfortunately a hopeless situation because of the possibility that, for instance, the coefficients of q lie in 1 Q Z, for some Q > L 2 . In this case, the values taken by q are 1 Q -separated and hence, for moderate values of Q, not likely to lie in any particular interval of length L −4r .…”

“…Summing over all < M 3 choices of indices gives an upper bound of M 3 (20ρ) D < 1 4 on the probability that (1) fails (since D is sufficiently large).…”

“…Pick a dimension D and consider the torus T D = R D /Z D . In this torus, consider a thin annulus, the projection π(A) of the set A = {x ∈ R D : 1 4 − N −4/D…”

“…If one then considers the progression P = {n 0 + nd : n √ N /10} (say), one sees that P ⊂ [N], and that θ 1 (n 0 + nd) − 1 2 T < 1 4 for all n √ N /10. That is, all points of P avoid the annulus π(A) (and in fact the whole ball of radius 1 4 ) since their first coordinates are confined to a narrow interval about 1 2 . Therefore P is coloured entirely red.…”