This paper provides a survey of methods, results, and open problems on graph and
hypergraph colourings, with a particular emphasis on semi-random “nibble” methods. We …

An independent set may not contain both a vertex and one of its neighbours. This basic fact
makes the uniform distribution over independent sets rather special. We consider the hard …

A graph $ G $ is $ k $-locally sparse if for each vertex $ v\in V (G) $, the subgraph induced
by its neighborhood contains at most $ k $ edges. Alon, Krivelevich, and Sudakov showed …

A $ k $-uniform hypergraph (or $ k $-graph) $ H=(V, E) $ is $ k $-partite if $ V $ can be
partitioned into $ k $ sets $ V_1,\ldots, V_k $ such that each edge in $ E $ contains precisely …

Suppose G and H are bipartite graphs and induces a partition of such that the subgraph of H
induced between and is a matching, whenever. We show for each that if H has maximum …

A seminal palette sparsification result of Assadi, Chen, and Khanna states that in every $ n $-
vertex graph of maximum degree $\Delta $, sampling $\Theta (\log n) $ colors per vertex …

List coloring is an influential and classic topic in graph theory. We initiate the study of a
natural strengthening of this problem, where instead of one list‐coloring, we seek many in …

Applications of graph colouring often involve taking restrictions into account, and it is
desirable to have multiple (disjoint) solutions. In the optimal case, where there is a partition …

Given a bipartite graph in which any vertex in (resp.,) has degree at most (resp.,), suppose
there is a partition of that is a refinement of the bipartition such that the parts in (resp.,) have …

This thesis deals with problems in what one can call extremal combinatorics, focused on
graph theory. Extremal graph theory mostly refers to a more specific subfield, which we will …