## New directions in additive combinatorics: day 4

The longer the conference goes, the more time I spend doing research with some of the participants, and I tend to day-dream more in the lecture, so the quality of reporting will inevitably be low. Ben Green completed his mini-course on finite field models in additive combinatorics, with many many applications of the Cauchy-Schwarz inequality. We then had a session of short talks by Peter Sin and Xiang-dong Hou. The former spoke on generalised adjacency matrices of graphs and when two such matrices of the same graph can be similar and be “Smith Normal Form” equivalent. The latter outlined a proof of a conjecture on “monomial” graphs, which has connections to generalised quadrangles (since the conjecture is about a girth 8 bipartite graph). After lunch, Peter Keevash finished off the proof of his fabulous theorem by wrapping up the strategy that he outlined in the first lecture. We then completed the day by a very nice session of short talks by Alice Hui and Sebastian Cioabă. The former gave a very nice result on switching strongly regular graphs arising from geometric configurations in symplectic spaces, and the latter gave a stimulating summary of the speaker’s work on different types of connectivity and expansion properties of distance-regular graphs and graphs coming from association schemes.

## New directions in additive combinatorics: day 3

Yesterday, we had four 90 minute talks! We began with more of the details of Peter Keevash’s proof, that uses some interesting results on hypergraphs and counting paths. After morning tea, Aart Blokhuis delved into t-fold blocking sets, followed by Simeon’s introduction to the links with coding theory. In the afternoon, we moved to the Department of Mathematics (at NUS) to see a colloquium by Ben Green on “Permutations and Number Theory”; a fabulous talk, one of the best I’ve ever seen him give. Finally, the “young person’s talk” was given by Ameera Chowdhury who spoke on a cool way to view and prove the MDS conjecture for prime fields and in the De Beule – Ball bound case.

In the evening, I spent two hours with Qing Xiang, Tao Feng, and Koji Momihara chatting about some interesting directions and problems we could look at in finite geometry. These guys are very good with Gauss sums and cyclotomic constructions, and so we are looking for more problems of this kind. Watch this space … work is now underway to construct some interesting objects in finite geometry! What was most interesting in our two hour session was the bottomless amount of notes and random pieces of paper that Qing seems to have stashed in his bag. Often I would be at the whiteboard saying something like “and from some work I did ten years ago …” and then Qing would pull out the relevant pages of information from his mystery bag. It is conjectured that he could also find a rabbit in there with some extra effort.

## New directions in additive combinatorics: day 2

I forgot to write the report last night as I got carried away with some mathematical discussions with a colleague; better late than never! First, I missed two talks today, due to forgetting the time mainly when I was talking with Simeon Ball about -arcs of projective spaces. We’ve ended up doing something, and that’s what has occupied me in the last while. Anyway, Aart Blokhuis and Simeon Ball began their mini-course on “Polynomial methods in finite geometry” yesterday, beginning with blocking sets. What I took away was that Hasse derivatives do something that the standard derivatives do not, but I’m still at a bit of loss why they are so successful in capturing the information about directions determined by a function. After morning tea, Lev spoke on the state of the art on quadratic residues and difference sets. I found this harder to follow, but there were some very interesting tables on computer output where some strange things happen. Five primes come out as solutions on a test of trillions of integers. Luckily we will have the slides posted on the webpage so I can remember exactly what these computations were about. Then Stefaan De Winter gave a beautiful talk on partial difference sets, where he and his co-authors have knocked off most of Ma’s list of parameters on at most 100 vertices. This was very impressive. There’s more on this talk over at Peter Cameron’s blog.

In the afternoon, Ben Green gave the second part of his series; this time on Rusza’s results and various improvements and advances thereof. Today (the third day) he will be giving a colloquium in the mathematics department here. As I said above, I lost track after afternoon tea and missed Ken Smith and Jim Davis’s talks; but then regained my composure and attended Oktay Olmez’s talk on directed strongly regular graphs and partial geometric designs; a great finish to another fantastic day at NUS!

## New directions in additive combinatorics: day 1

I know I’m not going to be able to keep this up, but I felt I needed to report on the first day of what is turning out to be an excellent conference at the National University of Singapore. This will leave me with the issue that my reporting of subsequent talks in this conference will not be at the same degree as I will present here, even though they might be of the same standard. So I apologise in advance.

First of all, we kicked off with Ben Green’s first lecture in his series on Finite Field Models (and in particular, to problems simulating outstanding problems in additive combinatorics). What was quite novel about this talk was that the news about the Sunflower Conjecture was placed in the centre of his talk: it was a news breaking talk. Add to this Tao’s reformulation of the Croot-Lev-Pach principle, and it really felt like everything we were being told was hot off the press. The next talk was by Peter Cameron, and he spoke about some problems to do with synchronisation (for automata and permutation groups). I was reminded that we still not have an idea what is going on with acting on *k*-subsets for . Something perhaps worth thinking about when I have time. Then Kai-Uwa Schmidt spoke about his recent results on *o-polynomials*; which are the functions you get when you coordinatise a hyperoval in a finite Desarguesian projective plane. I remember seeing his papers on the arxiv some time ago, so it was nice to see it all placed in context: some very nice recent results there on a problem that hasn’t seen many advances for a while. There were two citations of SymOmega in his talk: one of them was a comment I made about hyperovals, the second was a comment from Tim Penttila.

In the afternoon, Peter Keevash set the scene for his series on designs. He gave his big result of yesteryear, and all of its many consequences: Wilson’s Conjecture and the Existence Conjecture. He has only just outlined the strategy behind the proof, so the best is yet to come. I then gave the first talk after afternoon tea: it was a tad rushed, but I managed to get to the end. After my talk was a summary of the EKR problem for finite polar spaces by Klaus Metsch. Klaus has produced many interesting results in this area of late, and often by applying the Hoffmann bound to linear combinations of adjacency matrices of an association scheme! He showed that good old finite geometry techniques pair up well with the algebraic combinatorial techniques, when both cannot do the job alone.

There might be more reporting tomorrow … if I have time.

## Postdoc position in Perth

Cai Heng Li, Gabriel Verret and myself are currently advertising for a two year postdoc to work on our ARC grant entitled `Symmetries of finite digraphs’. The deadline is the 19th of February. More details are available at

http://external.jobs.uwa.edu.au/cw/en/job/496223/research-associate-ref-496223

## Minion and hamiltonicity

Over the last few years, Minion (I’m giving the link because if you try to search for it, you’ll be swamped by *Despicable Me* merchandise) has become one of my “go-to” tools, mostly because for certain specific types of searches it seems to perform just as well as, or even better than, a bespoke program. And of course, writing, debugging and optimising a bespoke program is enormously time-consuming, while writing a Minion model is usually very quick.

Most recently, I had an email from Brendan (McKay), who has found three planar cubic graphs of girth 5 that are *hypohamiltonian* – a graph is hypohamiltonian if it is not hamiltonian, but for every vertex , the graph is hamiltonian. Brendan’s graphs had 76 vertices, so not huge, but big enough.

## The shameful conjecture

This is the story of the ** shameful conjecture**, which started with Dominic Welsh, was almost solved by Paul Seymour, and eventually resolved by my friend Fengming Dong from Nanyang in Singapore.

The conjecture concerns a graph parameter called the *mean colour number* defined as follows: If is an -vertex graph, then is the average number of colours used over all proper colourings of using a palette of n colours.

As this is a slightly strange parameter, an example is probably in order, and it seems only appropriate to use the ** Welsh graph** (that is, the graph obtained by deleting one edge from ) for this purpose.

Using a palette of colours, there are colourings that use all four of the colours. There are colourings that use a fixed set of colours (the two vertices of degree two must be assigned the same colour), and there are ways of choosing colours of out , making a total of colourings using three colours. So the mean colour number of the Welsh graph is as the colourings are evenly divided between – and -colourings (there are no colourings with fewer than colours).

Using some straightforward identities, it can be shown that

where is the chromatic polynomial of .

The chromatic polynomial of the Welsh graph is so for this graph the expression has the value as expected.

It is obvious that the complete graph has the largest mean colour number, because every colouring requires colours, so .

At the other extreme is the empty graph on vertices, which has chromatic polynomial and thus

and as gets large the factor famously converges to from below.

But hang on, why did I say “the other extreme”? How do I ** know** that the empty graph has the lowest mean colour number? Of course, it seems intuitively obvious that adding an edge to a graph can only

*increase*the average number of colours required to colour it. Unfortunately, while it is intuitively obvious, it is not actually

*true*that adding edges increases the mean colour number. Mike Mosca (yes, the quantum crypto guy from Waterloo) found an infinite family of counterexamples, of which the smallest has only 6 vertices.

So is it even true that is the extreme graph? Bartels and Welsh conjectured that this is indeed the case – namely that for any -vertex graph

For a graph theorist, it is a little embarrassing to define a parameter applicable to all graphs, with the complete graph trivially at one extreme, and then be unable to show that the empty graph is the other extreme. In fact, more than just embarrassing, perhaps even positively ** shameful**.

Thus the shameful conjecture was christened.

Now Paul Seymour enters the ring. Normally if he turns his attention to some topic, even briefly, he quickly vacuums up and solves most of the known unsolved conjectures, generates some much harder and deeper conjectures, and then moves on. In this case though, he didn’t *quite* knock off the shameful conjecture. Instead, he proved that for any n-vertex graph ,

As , this value is just a little bit *too low* to prove the shameful conjecture.

The shameful conjecture hung on for another few years, but ultimately Fengming Dong (who has resolved quite a number of difficult chromatic and flow polynomial questions), used one of his typically intricate and ingenious proofs to show that the shameful conjecture is indeed true.

I don’t know if the shameful conjecture has stimulated any further research, or if the mean colour number has found any other uses, but I’ve always liked the story…

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