This volume contains eight survey articles by the invited speakers of the 29th British Combinatorial Conference, held at Lancaster University in July 2022. Each article provides an overview of recent developments in a current hot research topic in combinatorics. These topics span graphs and hypergraphs, Latin squares, linear programming, finite fields, extremal combinatorics, Ramsey theory, graph minors and tropical geometry. The authors are among the world's foremost researchers on their respective topics but their surveys are aimed at nonspecialist readers: they are written clearly with little prior knowledge assumed and with pointers to the wider literature. Taken together these surveys give a snapshot of the research frontier in contemporary combinatorics, making the latest developments accessible to researchers and graduate students in mathematics and theoretical computer science with an interest in combinatorics and helping them to keep abreast of the field.

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Preface Anthony Nixon and Sean Prendiville 1. Fair partitions Noga Alon 2. Hypergraph Turán Problems in \ell_2-norm József Balogh, Felix Christian Clemen and Bernard Lidický 3. Circuit imbalance measures and linear programming Farbod Ekbatani, Bento Natura and László A. Végh 4. Intersection problems in extremal combinatorics: theorems, techniques and questions old and new David Ellis 5. Finite geometry and extremal graph theory Valentina Pepe 6. Rainbow subgraphs and their applications Alexey Pokrovskiy 7. Explicit bounds for graph minors Paul Wollan 8. Convex and combinatorial tropical geometry Josephine Yu.

Anthony Nixon, Lancaster UniversityAnthony Nixon is Senior Lecturer in Mathematics at Lancaster University.

Sean Prendiville, Lancaster UniversitySean Prendiville is Lecturer in Mathematics at Lancaster University.

Anthony Nixon, Sean Prendiville, Noga Alon, József Balogh, Felix Christian Clemen, Bernard Lidický, Farbod Ekbatani, Bento Natura, László A. Végh, David Ellis, Valentina Pepe, Alexey Pokrovskiy, Paul Wollan, Josephine Yu