Extremal Combinatorics

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My research will focus on Extremal Combinatorics. A typical problem in Extremal Set Theory is, given an integer m and a property P, to find the largest size of a family of subsets of {1,2,..,m} which satisfy property P. One can also seek structural or enumerative information. The remarkable extremal graph results of Erdos, Stone and Simonovits provide examples. One seeks asymptotic bounds on the number of edges in a (simple) graph G on m vertices with no subgraph H. The asymptotic bounds are based on the chromatic number of H. There are a number of ways to generalize to set system, where we can interpret the sets as edges. We say a (0,1)-matrix is simple if it has no repeated columns.We encode a set system as a simple (0,1)-matrix using the vertex-set incidence matrix A. A subgraph generalizes as follows. Let F be an rxs (0,1)-matrix. We say F is a configuration of A if there is an rxs submatrix of A which is a row and column permutation of F. We consider forbidden configurations. An important example is Kk, the kx2k matrix of all (0,1)-columns with k rows. Vapnik and Chervonenkis studied this for Computational Learning Theory and it remains important. The VC-dimension is the largest value of k so that A contains a copy of Kk. Anstee and Sali conjectured that the maximum number of columns in a simple m-rowed matrix A that has no configuration F can be determined asymptotically (as a function of m) from a certain restricted range of product constructions. We will pursue this conjecture. Proof techniques include linear algebra/polynomial methods, clever inductions, Stability results, Shifting, Genetic algorithms and computer aided case analysis. Related problems include forbidden patterns, Sperner Theory, extremal graph theory, Ramsey Theory, Block designs, covering arrays, and block designs. With Sali, some best possible bounds were found when forbidding a configuration F whose size is a function of m. We also consider the property of a forbidden submatrix aiming for the conjecture of Anstee, Frankl, Furedi and Pach on Forbidden submatrices. Amortized analysis (from Computer Science) is used. I will also investigate problems in Graph Theory. A sample problem is the 2mx2m checkerboard. Can we cover it in dominoes? These correspond to perfect matchings in the grid graph. What happens if we delete some squares? Or if we fix the placement of some dominoes? This has natural extensions to grid graphs in higher dimension where the 2mx2mx2m can serve to model molecular structure. We found asymptotically best possible results for vertex deletion. A related problem is decomposing the edges of a graph into subgraphs with certain properties. Decomposing a graph into matchings is the edge colouring problem. I exploit the idea that looking for fractional subgraphs (that is, allowing an edge to be chosen with a fractional weight) is a network flow problem. This yields fast algorithms and existence theorems and new perspectives to be explored.