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In this paper we solve, by computational means, an open problem of Erickson: denoting [n] = {1,. .. , n}, what is the smallest integer n 0 such that, for every n ≥ n 0 and every 2-coloring of the grid [n] × [n], there is a constant 2-square, i.e. a 2 × 2 subgrid S = {i, i + t} × {j, j + t} whose four points are colored the same? It has been shown recently… (More)

Nested Monte-Carlo Search (NMC) and Nested Rollout Policy Adaptation (NRPA) are Monte-Carlo tree search algorithms that have proved their efficiency at solving one-player game problems, such as morpion solitaire or sudoku 16x16, showing that these heuristics could potentially be applied to constraint problems. In the field of Ramsey theory, the weak Schur… (More)

Eliahou, S., The 3x+ 1 problem: new lower bounds on nontrivial cycle lengths, Discrete Mathematics 118 (1993) 45556. Let 7': N-+ N be the function defined by T(n) = n/2 if n is even, T(n) = (3n + 1)/2 if n is odd. We show, among other things, that any nontrivial cyclic orbit under iteration of T must contain at least 17 087 915 elements.

Let D n be the dihedral group of order 2n. For all integers r, s such that 1 ≤ r, s ≤ 2n, we give an explicit upper bound for the minimal size µ D n (r, s) = min |A · B| of sumsets (product sets) A · B, where A and B range over all subsets of D n of cardinality r and s respectively. It is shown by construction that µ D n (r, s) is bounded above by the known… (More)

Steinhaus graphs on n vertices are certain simple graphs in bijec-tive correspondence with binary {0,1}-sequences of length n − 1. A conjecture of Dymacek in 1979 states that the only nontrivial regular Steinhaus graphs are those corresponding to the periodic binary sequences 110...110 of any length n − 1 = 3m. By an exhaustive search the conjecture was… (More)