On finite pattern-free sets of integers

Lutz G. Lucht

Techn. univ. Clausthal

Abstract

For n in N let the set A subset {1, ..., n} have the property that it does not contain any solution x, y, z to the equation sigma: az+by=cz, where a, b, c (c differs from a+b) are fixed positive integers. Such a set A is called sigma-free. In the case of sum-free sets, i.e., for a=b=1 and c different from 2, Ruzsa (1995) asked for the upper maximal density bound

D(sigma):=limsup(max{|A|: A subset {1, ..., n} sigma-free}/n).

Baltz, Hegarty, Knape, Larsson and Schoen (2005) determined this bound, which requires both a construction and a proof of its optimality. The latter is lengthy and intricate. The talk explains a different strategy of proof applicable not only to sum-free sets.