On sets without k-term arithmetic progression

摘要

For positive integers n and k, let rk(n) be the size of the largest subset of {1,2,…,n} without arithmetic progressions of length k. The van der Waerden number W(k1,k2,…,kr) is the smallest integer w such that every r-coloring of {1,2,…,w} contains a monochromatic ki-term arithmetic progression with color i for some i. In this note, an algorithm is proposed to search exact values of rk(n) for some k and n, and some new exact values of rk(n) for k=4,5,6,7,8 are obtained. The results extend the previous ones significantly. It is also shown that rk+1(2k2+1)⩾2k2−3k+3 for prime k⩾3, and three lower bounds for van der Waerden numbers are given: W(3,4,5)⩾124, W(5,8)⩾248, W(5,9)⩾320.