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.