A proof of the Ryser-Brualdi-Stein conjecture for large even nn

A Latin square of order $n$ is an $n$ by $n$ grid filled using $n$ symbols so that each symbol appears exactly once in each row and column. A transversal in a Latin square is a collection of cells which share no symbol, row or column. The Ryser-Brualdi-Stein conjecture, with origins from 1967, states that every Latin square of order $n$ contains a transversal with $n-1$ cells, and a transversal with $n$ cells if $n$ is odd. Keevash, Pokrovskiy, Sudakov and Yepremyan recently improved the long-standing best known bounds towards this conjecture by showing that every Latin square of order $n$ has a transversal with $n-O(\log n/\log\log n)$ cells. Here, we show, for sufficiently large $n$, that every Latin square of order $n$ has a transversal with $n-1$ cells. We also apply our methods to show that, for sufficiently large $n$, every Steiner triple system of order $n$ has a matching containing at least $(n-4)/3$ edges. This improves a recent result of Keevash, Pokrovskiy, Sudakov and Yepremyan, who found such matchings with $n/3-O(\log n/\log\log n)$ edges, and proves a conjecture of Brouwer from 1981 for large $n$.Comment: 71 pages, 13 figure

Multicoloured Random Graphs: Constructions and Symmetry

This is a research monograph on constructions of and group actions on countable homogeneous graphs, concentrating particularly on the simple random graph and its edge-coloured variants. We study various aspects of the graphs, but the emphasis is on understanding those groups that are supported by these graphs together with links with other structures such as lattices, topologies and filters, rings and algebras, metric spaces, sets and models, Moufang loops and monoids. The large amount of background material included serves as an introduction to the theories that are used to produce the new results. The large number of references should help in making this a resource for anyone interested in beginning research in this or allied fields.Comment: Index added in v2. This is the first of 3 documents; the other 2 will appear in physic

An extensive English language bibliography on graph theory and its applications

Bibliography on graph theory and its application