Local approaches to global problems in extremal combinatorics

Coulson, Matthew (2020). Local approaches to global problems in extremal combinatorics. University of Birmingham. Ph.D.

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Abstract

In this thesis we consider five problems in extremal combinatorics all of which which are all amenable to approaches based on local structure.

The first part of this thesis looks at rainbow subgraphs at extremal thresholds. We show that as soon as they appear, we can also find rainbow copies of Perfect Matchings, H-factors and Hamilton cycles in large graphs.

We then look to random digraphs and consider the D(n, p) model in which each edge is present independently with probability p. We find tail bounds on the size of the largest strongly connected component in the critical window around p = 1/n.

Finally, we consider the partition function of the ferromagnetic Potts model on graphs of bounded maximum degree. We show that there exists an open set in C containing an interval [1, w] inside which the partition function has no zeros.

Type of Work:

Thesis (Doctorates > Ph.D.)

Award Type:

Doctorates > Ph.D.

Supervisor(s):