A set X is said to be m-convex , m integer >=2, if for each set of m points at least one of the associated line segments lies in X. A 3-convex set is sometimes also known as Valentine convex after F.A.Valentine’s seminal 1957 paper : see H.G. Eggleston’s 1976 paper on the subject. The planar stars on the USA and EU flags are examples of planar 3-convex sets which are not unions of 2 convex sets. Clearly Ramsey’s theorem gives a general bound for the intersection of two 3-convex sets i.e.6. However in R2 and R3 the result is not best possible for the intersection of two , compact 3-convex sets : 5 is. While in R4 6 is indeed best possible . I have very inelegant solutions to the 2 and 3 dimensional cases some of which go back to my 1978 Ph.D thesis and some of which are recent. The 4 dimensional case follows from a minor modification of the construction in Eggleston’s 1976 paper. My question is a request for elegant solutions, or recent references, to these problems and their natural generalisations.
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$\begingroup$ In what sens the intersection of two sets is $6$ or $5$? $\endgroup$– Alex DegtyarevCommented Jun 13, 2016 at 11:50
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$\begingroup$ The intersection of two 3-convex sets, X and Y, is 6 convex i.e. for every six points in the set intersection at least one of fifteen associated line segments lies in both sets , by Ramsey's theorem. $\endgroup$– Ian CalvertCommented Jun 13, 2016 at 11:59
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