Schur numbers

A set X of integers is called sum-free if for every elements x and y from X, x+y is not in S (x and y do not need to be different).  The largest integer n such that the set {1,2,...,n} can be partitioned into k sum-free sets is called the Schur number S(k). The problem is to decide whether an integer n satisfies n <= S(k).

INPUT: A set of ground atoms "part(i)", i=1,2,...,k, where k is the number of parts and "number(i)", i=1,2,...,n, where n is the size of the set to partition.

OUTPUT: A partition of {1,2,...,n} into k sum-free parts. If one can be found, n <= S(k). Otherwise, n > S(k).


  1. S(4)=44. S(5) is not known. It is definitely at least 160 possibly, exactly 160
  2. Reasonable values for k and n: k =4, n=43, 44, 45; k=5, n=100, 110, 120, 130, 140