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# Extremal Combinatorics: existence, counting and typical structure

## Periodic Reporting for period 3 - ExtComb (Extremal Combinatorics: existence, counting and typical structure)

Reporting period: 2022-01-01 to 2023-06-30

The main project area is probabilistic and extremal combinatorics, with applications to algorithms. This is a rapidly expanding field of Mathematics, which has connections to many areas, e.g. probability, analysis, number theory, theoretical computer science and statistical physics.

Indeed, the study of combinatorial structures such as graphs provides the theoretical foundation for the analysis of many large networks arising in theoretical computer science, scheduling and communication. In the simplest case, these networks give rise to a graph, which consists of vertices, with suitable pairs of these vertices connected by edges. Hypergraphs arise when modelling non-binary relationships: instead of pairs we can also connect triples or larger vertex sets by a single hyperedge. Hypergraphs turn out to be much more challenging to investigate than graphs, so that much less is known about their properties.

A central theme of extremal combinatorics is the interplay and relationship between the parameters of combinatorial objects. The first and most immediate question which arises in this context is that of the (i) existence of objects with a given set of parameters. Once this has been answered, the next natural step is to ask for (ii) the number of such objects - i.e.to ask for a counting result. This is of fundamental importance in the context of many combinatorial questions arising in statistical physics. A very fruitful approach here is to ask for asymptotic results rather than exact formulas. This is sufficient for many applications and allows for the use of powerful techniques e.g. from probabilistic combinatorics. Moreover, this asymptotic approach sometimes makes it possible to go even further and ultimately uncover the (iii) typical structure of the objects in such a given class.

In this project, we consider the above perspective with a focus on inter-related topics involving combinatorial designs, decompositions, Latin squares as well as matchings and colourings in graphs and hypergraphs.
These include longstanding open problems of central importance as well as promising new directions that are currently emerging. Moreover, the project themes have close connections e.g. to probability and theoretical computer science.
The main focus of the project is on extremal and probabilistic combinatorics, with applications to algorithms and theoretical computer science. The main objects of interest are graphs and hypergraphs: these consist of a set of vertices, some of which are connected by (hyper-)edges.

A very recent highlight is the proof of the Erdos-Faber-Lovasz conjecture. This famous conjecture dates back to 1972 and concerns colourings of locally sparse hypergraphs (more precisely, the aim is to colour the edges of any n-vertex linear hypergraph with n colours so that edges of the same colour are pairwise disjoint). Such colouring problems are central to combinatorics, and have applications e.g. to scheduling problems and geometry. Erdos considered this as one of his three favourite problems in combinatorics. As the difficulty of the conjecture became apparent, he offered successively increasing rewards for a proof of the conjecture, which eventually reached \$500, one of his highest prizes. We were able to prove this for all large hypergraphs based on probabilistic techniques, with tools involving decomposition results developed earlier by some of the team members.

Another highlight is our resolution of a 40-year old conjecture on Hamilton cycles in random subgraphs of the hypercube. The hypercube and its subgraphs are central objects in graph theory and computer science, e.g. as a sparse network model with strong connectivity properties. We determine the threshold for a binomial random subgraph of the hypercube with edge probability p to be Hamiltonian (i.e. to obtain a cycle using all the vertices). In fact, we obtain a precise `hitting time’ version of this result.

Results within the project also include:
- a proof of a classical conjecture of Alspach, Mason and Pullman (from 1976) on path decompositions of tournaments
- a proof of conjectures of Brualdi-Hollingsworth as well as Constantine on decompositions of edge-coloured complete graphs into rainbow spanning trees
- an improvement of a 20-year old bound on a classical problem of finding large matchings in regular hypergraphs
The project will continue further along the themes outlined in the description of work, i.e. designs & decompositions, Latin squares as well as matchings in graphs and hypergraphs.
In addition, the project will explore further applications and extensions of techniques developed for such problems to other areas, in particular to graph and hypergraph colouring problems.