On sets without k-term arithmetic progression

Abstract

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