## Problems and results on linear hypergraphs

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## Abstract

In this thesis, we tackle several problems involving the study of 3-uniform, linear hypergraphs satisfying some additional structural constraint.

We begin with a problem of Hrushovski concerning Latin squares satisfying a partial associativity condition. From an `99$\%$' version and then to a `

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We then take this problem further. A condition due to Thomsen provides a combinatorial constraint which, if satisfied by the Latin square

We also study a problem of Loh concerning sequences of triples of integers from

Finally, we present a collection of shorter results. In work connecting to the earlier chapters, we resolve the Brown--Erd\H os--S'os conjecture in the context of hypergraphs with a group structure, and show moreover that subsets of group multiplication tables exhibit local density far beyond what can be hoped for in general. In work less closely connected to the main theme of the thesis, we also answer a question of Leader, Mili'cevi'c and Tan concerning partitions of boxes, consider a problem on projective cubes in

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EPSRC (1650384)