## Chromatic number of the plane

The unit distance graph on has edges between those pairs of points at Euclidean distance .  The chromatic number of this graph lies between (by exhibiting a small subgraph on vertices with chromatic number ) and (by an explicit colouring based on a hexagonal tiling of the plane).  Aubrey de Grey has just posted a …

## Random Structures and Algorithms 2017

A partial, chronologically ordered, list of talks I attended at RSA in Gniezno, Poland. Under construction until the set of things I can remember equals the set of things I’ve written about. Shagnik Das A family of subsets of that shatters a -set has at least elements. How many -sets can we shatter with a …

## Clique decompositions of multipartite graphs and completion of Latin squares

Ben Barber, Daniela Kühn, Allan Lo, Deryk Osthus and Amelia Taylor Journal of Combinatorial Theory, Series A, Volume 151, October 2017, Pages 146–201 PDF A Latin square of order is an grid of cells, each of which contains one of distinct symbols, such that no symbol appears twice in any row or column.  There is a natural …

## Matchings without Hall’s theorem

In practice matchings are found not by following the proof of Hall’s theorem but by starting with some matching and improving it by finding augmenting paths.  Given a matching in a bipartite graph on vertex classes and , an augmenting path is a path from to such that ever other edge of is an edge of …

## Matchings and minimum degree

A Tale of Two Halls (Philip) Hall’s theorem.  Let be a bipartite graph on vertex classes , .  Suppose that,  for every , .  Then there is a matching from to . This is traditionally called Hall’s marriage theorem.  The picture is that the people in are all prepared to marry some subset of the …

## Partition regularity and other combinatorial problems

This is the imaginative title of my PhD thesis.  It contains four unrelated pieces of work.  (I was warned off using this phrasing in the thesis itself, where the chapters are instead described as “self-contained”.) The first and most substantial concerns partition regularity.  It is a coherent presentation of all of the material from Partition …

## Nowhere zero 6-flows

A flow on a graph is an assignment to each edge of of a direction and a non-negative integer (the flow in that edge) such that the flows into and out of each vertex agree.  A flow is nowhere zero if every edge is carrying a positive flow and (confusingly) it is a -flow if the flows …

## Fractional clique decompositions of dense graphs and hypergraphs

Ben Barber, Daniela Kühn, Allan Lo, Richard Montgomery and Deryk Osthus, Journal of Combinatorial Theory, Series B, Volume 127, November 2017, Pages 148–186 PDF Together with Daniela Kühn, Allan Lo and Deryk Osthus I proved that for every graph there is a constant such that every “-divisible” graph on vertices with minimum degree at least has an …

## Edge-decompositions of graphs with high minimum degree

Ben Barber, Daniela Kühn, Allan Lo and Deryk Osthus, Advances in Mathematics, Volume 288, 22 January 2016, Pages 337–385 PDF When can the edge set of a graph be partitioned into triangles? Two obvious necessary conditions are that the total number of edges is divisible by 3 and the degree of every vertex is even. We …

## Random walks on quasirandom graphs

Ben Barber and Eoin Long, The Electronic Journal of Combinatorics, 20(4) (2013), #P25 PDF Take a long (proportional to ) random walk in a quasirandom graph . Must the subgraph of edges traversed by be quasirandom? We’d like to say yes, for the following reason: visits every vertex about the same number of times, so we …