On Schur properties of random subsets of integers

[摘要] A classic result of I. Schur [9] asserts that for every r greater than or equal to 2 and for n sufficiently large, if the set [n] = {1, 2, ..., n} is partitioned into r classes, then at least one of the classes contains a solution to the equation x + y = z. Any such solution with x not equal y will be called a Schur triple. Let us say that A subset of or equal to [n] has the Schur property if for every partition (or 2-coloring) of A = R boolean OR B (for red and blue), there must always be formed some monochromatic Schur triple, i.e., belonging entirely to either R or B. Our goal here is to show that n(1/2) is a threshold for the Schur property in the following sense: for any omega = omega(n) --> infinity, almost all sets A subset of or equal to[n] with \A\ < n(1/2)/omega do not posses the Schur property, while almost all A subset of or equal to[n] with \A\ > omega n(1/2) do. (C) 1996 Academic Press, Inc.

[发布日期] 1996-12-01