A note on convex subsets of

Abstract: R. Graham, M. Simonovits, and V.T. Sós [R. Graham, M. Simonovits, V.T. Sós, A note on the intersection properties of subsets of integers, J. Combin. Theory, Ser. A 28 (1980) 106–110] in 1980 posed the following question. Suppose is a convex subset of and let be a family of subsets of such that is convex and nonempty for . If is a family of maximum cardinality with this property, is it true that is also non-empty? In this paper, we answer this question in the negative by exhibiting a set, , and a family of subsets of for every dimension . [Copyright &y& Elsevier]