COMBINATORIAL OPTIMIZATION APPROACHES TO DISCRETE PROBLEMS

As stressed by the Society for Industrial and Applied Mathematics (SIAM): Applied mathematics, in partnership with computational science, is essential in solving many real-world problems. Combinatorial optimization focuses on problems arising from discrete structures such as graphs and polyhedra. This thesis deals with extremal graphs and strings and focuses on two problems: the Erdős’ problem on multiplicities of complete subgraphs and the maximum number of distinct squares in a string. The first part of the thesis deals with strengthening the bounds for the minimum proportion of monochromatic t cliques and t cocliques for all 2-colourings of the edges of the complete graph on n vertices. Denote by kt(G) the number of cliques of order t in a graph G. Let kt(n) = min{kt(G) +kt(G)} where G denotes the complement of G of order n. Let ct(n) = kt(n)/ ( n t ) and ct be the limit of ct(n) for n going to infinity. A 1962 conjecture of Erdős stating that ct = 2 1− ( t 2 ) was disproved by Thomason in 1989 for all t ≥ 4. Tighter counterexamples have been constructed by Jagger, Sťovicek and Thomason in 1996, by Thomason for t ≤ 6 in 1997, and by Franek for t = 6 in 2002. We present a computational framework to investigate tighter upper bounds for small t yielding the following improved upper bounds for t = 6, 7 and 8: c6 ≤ 0.7445×2 ( 6 2 ) , c7 ≤ 0.6869 × 2 ( 7 2 ) , and c8 ≤ 0.7002 × 2 ( 8 2 ) . The constructions are based on a large but highly regular variant of Cayley graphs for which the number of cliques and cocliques can be expressed in closed form. Considering the quantity et = 2 ( t 2 ) ct, the

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