### Forced differences between terms of subsequences of integer sequences

Michael Gilpin, Robert Shelton
1983 Proceedings of the American Mathematical Society
Let a" a2,. ■ ■ be a sequence of integers and let D = {d¡.áj be a fixed finite set of integers. For each positive integer n we investigate the problem of choosing maximal subsequences a¡,... ,a¡ from a,.a" such that \a,^ -a. \ CO for a ¥= ß. An asymptotic form for t, the maximum length of such subsequences, is derived in the special case a, = i. 0. Introduction. We examine the problem of constructing subsequences a¡,... ,a¡ of a finite sequence ax,...,an of integers with the property that the
more » ... fferences j a, -a¡ | avoid a fixed set D of integers. The problem is illustrated by three examples in §1. In §2 a partial solution of the problem is stated and proved using the pigeonhole principle. In §3 we define, and give some elementary properties of the function L({a,}f, D: n), where L({a¡}f, D: n) is the length of the longest subsequence a"... ,a¡ that can be chosen from ax,... ,an so that the differences | a: -a¡ \ are never in D. In §4 we prove that for the sequence 1, 2, 3,... of positive integers, the function L is essentially cyclic, and we investigate its asymptotic behavior.