Kemnitz, Arnfried On a lattice point problem. (English) Zbl 0539.05008 Ars Comb. 16-B, 151-160 (1983). 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 Cited in 3 ReviewsCited in 30 Documents MSC: 05D10 Ramsey theory 05B99 Designs and configurations 05C55 Generalized Ramsey theory Keywords:lattice points; centroid; generalized Ramsey theorem PDFBibTeX XMLCite \textit{A. Kemnitz}, Ars Comb. 16-B, 151--160 (1983; Zbl 0539.05008)