Predicates whose maximal length functions increase periodically

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Department of Mathematical Sciences


Let P be a predicate defined on finite sets of positive integers and define LP(n) to be the Largest cardinality of subsets of {1, 2, ..., n} for which P is false. We exhibit conditions on P which force the existence of integers N, M, and K so that LP(n + M) = LP(n) + K whenever n> N. For such predicates we say that LP increases periodically. In particular, we show that if D is a finite set of tuples of positive integers, then LP increases periodically for the predicate P="X contains as s-tuple {ai}s1 with {ai + 1 - ai}s - 11 in D". This extends the main result in [2], as well as results of Erdo{combining double acute accent}s, Hemminger and McKay, Liu, and Wagstaff.

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Discrete Mathematics