The combinatorial structure of (m,n)-convex sets

LetS be a closed subset of a Hausdorff linear topological space,S having no isolated points, and letc _s (m) denote the largest integern for whichS is (m,n)-convex. Ifc _s (k)=0 andc _s (k+1)=1, then $$ c_s left( m right) = sumlimits_{i = 1}^k {left( {begin{array}{*{20}c} {left[ {frac{{m + k - i}} {k}} right]} 2 end{array} } right)} $$ . Moreover, ifT is a minimalm subset ofS, the combinatorial structure ofT is revealed.