Periodic Reporting for period 4 - CC (Combinatorial Construction) Período documentado: 2020-07-01 hasta 2021-06-30 Resumen del contexto y de los objetivos generales del proyecto Combinatorial Construction is a mathematical challenge with many applications. Examples include the construction of networks that are very sparse but highly connected, or codes that can correct many transmission errors with little overhead in communication costs. For a general class of combinatorial objects, and some desirable property, the fundamental question in Combinatorial Construction is to demonstrate the existence of an object with the property, preferably via an explicit algorithmic construction. Thus it is ubiquitous in Computer Science, including applications to expanders, sorting networks, distributed communication, data storage, codes, cryptography and derandomisation. In popular culture it appears as the unsolved "lottery problem" of determining the minimum number of tickets that guarantee a prize. In 2014 I proved the Existence Conjecture for combinatorial designs, via a new method of Randomised Algebraic Constructions; this result has already attracted considerable attention in the mathematical community. The significance is not only in the solution of a problem posed by Steiner in 1852, but also in the discovery of a powerful new method, that promises to have many further applications in Combinatorics, and more widely in Mathematics and Theoretical Computer Science. This project has achieved its aim of developing this idea into a general method, thus resolving several other related longstanding open problems. Trabajo realizado desde el comienzo del proyecto hasta el final del período abarcado por el informe y los principales resultados hasta la fecha Significant progress has been achieved on all four objectives of the project, resulting in several publications which have been disseminated by team members at academic conferences. The first objective is to develop the method of Randomised Algebraic Construction. This has been achieved by two papers, one refining and simplifying my original proof of the Existence Conjecture, and the second greatly generalising it to encompass many new applications, including generalised forms of two conjectures of Ringel: the Oberwolfach Problem and his tree packing conjecture. The second paper also makes progress on the second objective, which is a unifying framework for the existence of perfect matchings in sparse graphs. Further progress on this objective has been achieved by a series of papers culminating in improved bounds for Ryser's Conjecture on transversals in latin squares. Within the third objective, which is the theory of random independent sets and matchings in hypergraphs, we have developed new techniques for approximations of partition functions. The fourth objective, a structural characterisation of isoperimetry in Hamming spaces, has been very substantially addressed by several papers, including stability results for the vertex-isoperimetric and edge-isoperimetric inequalities. Avances que van más allá del estado de la técnica e impacto potencial esperado (incluida la repercusión socioeconómica y las implicaciones sociales más amplias del proyecto hasta la fecha) In each of the four objectives, the work described above goes beyond the previous state of the art. An octahedral move for triangle decompositions
