A note on convex subsets of

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 S is a convex subset of Z k and let A be a family of subsets of S such that A i ∩ A j is convex and nonempty for A i , A j ∈ A , A i ≠ A j . If A is a family of maximum cardinality with this property, is it true that ∩ A i ∈ A A i is also non-empty? In this paper, we answer this question in the negative by exhibiting a set, S , and a family of subsets of S for every dimension k ≥ 2 .