## Diffy naive

A week or so ago Rob Eastaway posted about the game Diffy on Twitter. Diffy begins with four numbers arranged around a cycle. Taking the (absolute values of) differences between adjacent pairs produces four more numbers around a cycle. If you start with positive integers then iterating this process eventually reaches . How many iterations …

## Concentration inequalities

In early 2019 I gave three lectures on concentration inequalities from a combinatorial perspective to the postgraduate reading group SPACE (Sum-Product, Additive-Combinatorics Etc.) at the University of Bristol. I prepared some very rough notes on what was covered. You might also be interested in a scan of my undergraduate lecture notes on the same topic.

## 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 …

## The number of maximal left-compressed intersecting families

A family of sets (subsets of of size ) is intersecting if every pair of sets in have a common element.  If then every pair of sets intersect, so can be as large as .  If then the Erdős–Ko–Rado theorem states that , which (up to relabelling of the ground set) is attained only by the …

## Counting colourings with containers

On the maximum number of integer colourings with forbidden monochromatic sums, Hong Liu, Maryam Sharifzadeh and Katherine Staden Maryam spoke about this paper at this week’s combinatorics seminar. The problem is as follows.  Let be the number of -colourings of a subset of with no monochromatic sum .  What is the maximum of over all ? …

## Linear programming duality

The conventional statement of linear programming duality is completely inscrutable. Prime: maximise subject to and . Dual: minimise subject to and . If either problem has a finite optimum then so does the other, and the optima agree. I do understand concrete examples.  Suppose we want to pack the maximum number vertex-disjoint copies of a graph …

## 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 …

## 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 …

## Block partitions of sequences

Let be a line segment of length broken into pieces of length at most . It’s easy to break into blocks (using the preexisting breakpoints) that differ in length by at most (break at the nearest available point to , etc.). In the case where each piece has length and the number of pieces isn’t divisible …