New results in extremal combinatorics

Extremal problems, in general, ask for the optimal size of certain finite objects when some restrictions are imposed. In extremal combinatorics, a major field in combinatorics, one studies how global properties guarantee the existence of local substructures, or equivalently, how avoiding local substructures poses a constraint on global quantities. In this thesis, we study the following three problems in combinatorics. The first problem is related to the Brown–Erdos–Sos conjecture, which can be reformulated as a problem of finding a dense substructure, namely a large number of triples, spanned by a given number of elements in a quasigroup. In the special case of finite groups, we show in Chapter 2 that in every dense set of triples, there exists a subset of m elements which spans 4m/3 (1 − o(1)) triples, as m tends to infinity, which is much higher than the conjectured amount m − 3 for a general quasigroup. Later in the chapter, we give an elementary proof that, in finite groups, the maximum number of triples spanned by m elements has order m². The second problem concerns planar polygons in the 3-space. The maximum possible number of polygons from a given number of points is controlled by their intersection properties. Two hexagons in the space are said to intersect badly if their intersection consists of at least one common vertex as well as an interior point. In Chapter 3, we show that the number of hexagons on n points in 3-space without bad intersections is o(n²), under a mild assumption that the hexagons are ‘fat’. The main tool we used is the triangle removal lemma. The last problem in this dissertation is about the sum-product conjecture of Erdos and Szemeredi. The sum-product estimate concerns the larger size of the sumset and productset in terms of the size of the set itself. In Chapter 4 we prove an estimate with exponent 4/3 for the ring of quaternions and a certain family of well-conditioned matrices, using the boundedness of the kissing numbers.