A note on visible islands
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Given a finite point set P in the plane, a subset S ⊆ P is called an island in P if conv(S) ∩ P = S. We say that S⊂ P is a visible island if the points in S are pairwise visible and S is an island in P. The famous Big-line Big-clique Conjecture states that for any k ≥ 3 and ℓ≥ 4, there is an integer n = n(k,ℓ), such that every finite set of at least n points in the plane contains ℓ collinear points or k pairwise visible points. In this paper, we show that this conjecture is false for visible islands, by replacing each point in a Horton set by a triple of collinear points. Hence, there are arbitrarily large finite point sets in the plane with no 4 collinear members and no visible island of size 13.
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