This area studies when a large host graph or hypergraph must contain a given spanning subgraph, such as a Hamilton cycle, a perfect tiling, a spanning surface, or a more general spanning structure. The central question is which easily checkable density conditions, such as minimum degree, force this structure to appear.

My work takes place mostly in the hypergraph setting, including tight and loose Hamilton cycles, combinatorial spheres and perfect tilings. A recurring theme is to develop notions of robustness that directly connect abstract necessary conditions that certify lower bounds with the existence of the desired object. Once such a framework is obtained, many concrete problems can be solved with ease.

This area studies how the edge set of a graph can be split into well-structured parts. Central questions concern the possibility of decomposing a (hyper)graph into rigid parts, or the optimal number of parts when the objects are more flexible. A typical problem from the latter line of research is to colour the edges of a graph with few colours so that no two conflicting edges receive the same colour. The guiding question is how far the trivial lower bounds, usually arising from the maximum degree or other local constraints, are from the true answer.

My work establishes improved bounds for related problems, including local list edge-colouring, simultaneous edge-colouring and linear arboricity. While my earlier work focuses on structural analysis involving parameters such as treewidth, my recent work has shifted more towards a probabilistic toolset, which allows these problems to be resolved in large systems.

This area studies the phenomenon that any colouring of a sufficiently large structure must contain a well-structured monochromatic part. A classic example is Ramsey's theorem, which in its simplest case tells us that among six people there is always a set of three mutual friends or three mutual strangers.

Much of my work concerns a partitioning variant of this theme. Given an edge-coloured graph, the goal is to cover its vertex set with as few monochromatic pieces, usually cycles and paths, as possible. The guiding question is how this number depends on the number of colours and whether it can be bounded independently of the size of the graph. The methods involved often combine tools from extremal combinatorics, such as the regularity lemma and absorption, with careful structural analysis.