ISSN:
1572-9273
Keywords:
06A10
;
05A05
;
05C55
;
zero-sum
;
monotone sequence
;
partially ordered set (posets)
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract Bialostocki proposed the following problem: Let n≥k≥2 be integers such that k|n. Let p(n, k) denote the least positive integer having the property that for every poset P, |P|≥p(n, k) and every Z k -coloring f: P → Z k there exists either a chain or an antichain A, |A|=n and ∑ a∈A f(a) ≡ 0 (modk). Estimate p(n, k). We prove that there exists a constant c(k), depends only on k, such that (n+k−2)2−c(k) ≤ p(n, k) ≤ (n+k−2)2+1. Another problem considered here is a 2-dimensional form of the monotone sequence theorem of Erdös and Szekeres. We prove that there exists a least positive integer f(n) such that every integral square matrix A of order f(n) contains a square submatrix B of order n, with all rows monotone sequences in the same direction and all columns monotone sequences in the same direction (direction means increasing or decreasing).
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF00383966
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