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On a lattice point problem. (English) Zbl 0539.05008

Let \(f(r,d)\) be the minimal number of lattice points in Euclidean \(d\)-space such that every set of \(f(r,d)\) lattice points contains some \(r\) points which have a centroid with integer coordinates. For \(d\geq 2\), let \(g(r,d)\) denote the minimal number of such arbitrary lattice points which in addition are pairwise incongruent mod \(r\) in at least one coordinate. The following results are proved (some already proved)
\((r-1)2^ d+1\leq f(r,d)\leq(r-1)r^ d+1, f(2^ k,d)=(2^ k-1)2^ d+1, f(r,1)=2r-1, 2p-1\leq g(p,2)\leq 4p-3, g(5,2)=9, g(7,2)=13, f(5,2)=17, f(7,2)=25.\) The existence of \(f(r,d)\) is a generalized Ramsey theorem which makes it very difficult to evaluate this function.
Reviewer: M.Cheema

MSC:

05D10 Ramsey theory
05B99 Designs and configurations
05C55 Generalized Ramsey theory
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