On the structure of graphs with forbidden induced substructures

One of the central goals in extremal combinatorics is to understand how the global structure of a combinatorial object, e.g. a graph, hypergraph or set system, is affected by local constraints.
In this thesis we are concerned with structural properties of graphs and hypergraphs which locally do not look like some type of forbidden induced pattern. Patterns can be single subgraphs, families of subgraphs, or in the multicolour version colourings or families of colourings of subgraphs.

Erdős and Szekeres's quantitative version of Ramsey's theorem asserts that in every $2$-edge-colouring of the complete graph on $n$ vertices there is a monochromatic clique on at least $\frac{1}{2}\log n$ vertices. The famous Erdős-Hajnal conjecture asserts that forbidding fixed colourings on subgraphs ensures much larger monochromatic cliques. The conjecture is open in general, though a few partial results are known. The first part of this thesis will be concerned with different variants of this conjecture: A bipartite variant, a multicolour variant, and an order-size variant for hypergraphs.

In the second part of this thesis we focus more on order-size pairs; an order-size pair $(n,e)$ is the family consisting of all graphs of order $n$ and size $e$, i.e. ... mehron $n$ vertices with $e$ edges. We consider order-size pairs in different settings: The graph setting, the bipartite setting and the hypergraph setting. In all these settings we investigate the existence of absolutely avoidable pairs, i.e. fixed pairs that are avoided by all order-size pairs with sufficiently large order, and also forcing densities of order-size pairs $(m,f)$, i.e. for $n$ approaching infinity, the limit superior of the fraction of all possible sizes $e$, such that the order-size pair $(n,e)$ does not avoid the pair $(m,f)$.