The solution sets of extremal problems in $H\sp 1$

Extremal problems for chromatic neighborhood sets

Extremal problems on planar point sets

of Ph.D. thesis

Extremal Problems on Finite Sets and Posets

The overarching theme of the thesis is the investigation of extremal problems involving forbidden partially ordered sets (posets). In particular, we will be concerned with the function La(n, P ), defined to be the maximum number of sets we can take in the Boolean lattice 2[n] without introducing the relations of a poset P as containment relations among the sets. This function plays an analogous...

Extremal Problems for Geometric Hypergraphs 1 Extremal Problems for Geometric

A geometric hypergraph H is a collection of i-dimensional simplices, called hyperedges or, simply, edges, induced by some (i + 1)-tuples of a vertex set V in general position in d-space. The topological structure of geometric graphs, i.e., the case d = 2; i = 1, has been studied extensively, and it proved to be instrumental for the solution of a wide range of problems in combinatorial and compu...

Traces of Finite Sets: Extremal Problems and Geometric Applications

Given a hypergraph H and a subset S of its vertices, the trace of H on S is defined as H|S = {E ∩ S : E ∈ H}. The Vapnik–Chervonenkis dimension (VC-dimension) of H is the size of the largest subset S for which H|S has 2|S| edges. Hypergraphs of small VC-dimension play a central role in many areas of statistics, discrete and computational geometry, and learning theory. We survey some of the most...