Code and data

• search.c
• 221000.dat
• plot5.ggb – Geogebra notebook
• geo5.sh – awk one-liner to format array to input parameters to Geogebra

Errata

Every time I wrote I should have written . This will be corrected in the final version.

The pigeonhole upper bound is perhaps better written , since it arises as , where is the totient function.

Quality of the data

After commenting, Tomás Oliveira e Silva sent a follow-up email prompting me to belatedly flesh out this post a little.

After computing the first 100000 values independently, he found one discrepancy. 221000.dat has

53829 1.035945566231253e-8 6045 19754 30497 38849

but Tomás has

   53829 0.0000000043404367309945397      0      0  15622  31244 31244

which agrees with the output of search.c

53829 4.3404373281240489e-09 0 15622 31244 31244

So what’s going on here? The answer necessarily includes the fact that search.c was not used to produce that line of 221000.dat. The history is that the data set and the search program developed together over time. The final version of search.c is (hopefully) better than the earliest versions in several ways, including a considered approach to avoiding zero sums rather than an optimistic one, and a change in the enumeration order to notice when we’re about to generate a lot of very long sums and skip over them. The very earliest searches weren’t even done in C; I used Haskell, which is my usual go to for combinatorics, but wasn’t the best fit for this naturally expressed brute force loop. In any case, I didn’t rerun the whole search using the final program because of the cost of producing the data. I’ve been at peace with the risk of errors because I was using the data to suggest patterns to investigate rather than relying on it for any proofs, but I should have been clearer about what I did.

Putting my archaeologist’s hat on, we can tell that the line in 221000.dat isn’t from the latest version of search.c due to the different output format. In fact, there’s a break between the two formats at .

119999 1.6533522139157386e-9 15853 43211 70569 86422
120000 3.4249738497730593e-10 20224 40448 77495 82953
120001 8.4593861872462649e-10 10962 39803 71356 79549
120002 1.4113956883942480e-10 19401 45243 71085 90486
120003 1.3630084760844623e-10 10265 44351 65134 85917
120004 5.3309237833505616e-10 9345 39651 69957 79302
120005 2.1493219522622943e-09 6408 39397 67265 79033

This means that I might have used Haskell as far as 120000 (I might also have used C with a different output format), but I didn’t use it any later. In fact the incorrect line matches the output of the Haskell search exactly, so the error probably comes from a bug in the Haskell. My first guess, given the correct form of the optimal configuration for 53829, is that I had an off by one error in an index preventing me from looking at , but after revisiting the source code I don’t think that’s it. As an early version of the guard against pointless searching I have the line

bestCompletions bestGuess a b = if length > 2.0 then [] else [t | t@(_,a,b,c,d) <- candidates''', not $ zero a b c d]

The error here is that the cutoff for “too long” needs to be 2 plus the shortest thing seen so far, not 2. (Not counting the triple-prime in a variable name as an error.) The correct configuration for 53829 is exactly a situation in which the sum of the triple has length just over 2, so the Haskell search couldn’t see it.

I’m going to generate the data from 1 to 120000 again in case there are more errors of this type.

I’ve previously written about the Namer-Claimer game.  I can now prove that the length of the game is with optimal play from each side, matching the greedy lower bound.  The upper bound makes use of randomness, but in a very controlled way.  Analysing a truly random strategy still seems like it will be very difficult.

The proof brings up a surprising connection to the Ramsey theory of Hilbert cubes.

In Edge decompositions of graphs with high minimum degree, Daniela Kühn, Allan Lo, Deryk Osthus and I proved that the edge sets of sufficiently dense graphs satisfying necessary divisibility conditions could be partitioned into copies of an arbitrary graph .  This result has since been generalised to other settings by various authors.  In this paper we present a simplified account of the latest version of the argument, specialised to the case where is a triangle.

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 correspondence between Latin squares of order and partitions of into triangles.  We identify the three vertex classes of with the rows, columns and symbols of the Latin square.  A triangle corresponds to the symbol appearing in the intersection of row and column of the Latin square.  Since each cell contains exactly one symbol, and each symbol appears exactly once in each row and each column, the triangles corresponding to a Latin square do indeed partition .

What about partitions of into ‘s?  Identify the vertex classes of with rows, columns, red symbols and  blue symbols.  Then a corresponds to a red symbol and a blue symbol in the intersection of row and column .  If we look at just the red symbols or just the blue symbols then we see a Latin square.  But we also have the extra property that each pair of red and blue symbol appears in exactly one cell of the grid.  Two Latin squares with this property are called orthogonal.  So pairs of orthgonal Latin squares of order correspond to decompositions of into ‘s.  More generally, a sequence of  mutually orthogonal Latin squares of order  corresponds to a partition of the complete -partite graph on vertex classes of size into ‘s.

Suppose now that we have a partial Latin square of order , that is, a partially filled in grid of cells obeying the rules for a Latin square.  Can it be completed to a Latin square?  In the early 1980s several researchers proved that the answer is yes provided at most cells have been filled in total.  This is best possible, as if we place ‘s and a single on the main diagonal there is no legal cell in which to place the th .  The same example shows that it is not enough to ask only that each row and each column contains only a small number of non-empty cells.  But what if each row, column and symbol has been used at most times?  Can we then complete to a Latin square?  Daykin and Häggkvist conjectured that we can, provided .

What does this mean on the graph side?  Let be a subgraph of obtained by deleting a set of edge-disjoint triangles such that no vertex is in more than triangles.  Then should have a triangle-decomposition if .  In this paper we prove that has a triangle-decomposition provided .

In fact we prove something more general.  The ‘s obtained in this way have the properties that (i) each vertex has the same number of neighbours in each other vertex class and (ii) each vertex sees at least a proportion of the vertices in each other class (a partite minimum degree condition).  We prove that all such graphs have triangle-decompositions when .

The proof is based on that of the similar result for non-partite graphs in Edge decompositions of graphs with high minimum degree.  However, it is not simply a translation of that proof to the partite setting.  In the partite case we not only have to ensure that all of our gadgets can be embedded in partite graphs, we must also take care to ensure a stronger notion of divisibility is preserved throughout our decomposition process.  This makes the proof extremely technical.

We also prove the analogous result for -decompositions of complete -partite graphs, with the less impressive bound .  The connection to mutually orthogonal Latin squares is more complicated for , as the partially filled in cells only correspond to ‘s in the case where each non-empty cell contains one of each symbol, but we still show that there is some such that if each row, column and coloured symbol is used at most times then partial mutually orthogonal Latin squares can be complete.

As for the non-partite case, the bounds on are currently limited by available fractional or approximate decomposition results.  Improvements to these would lead automatically to improvements of the bounds in this paper.

The first and most substantial concerns partition regularity.  It is a coherent presentation of all of the material from Partition regularity in the rationals, Partition regularity with congruence conditions and Partition regularity of a system of De and Hindman.

The remaining three chapters are expanded versions of Maximum hitting for sufficiently large, Random walks on quasirandom graphs and A note on balanced independent sets in the cube.¹

Erratum: the form of Talagrand’s inequality quoted as Theorem 6 in Chapter 4 has an incorrect proof in the book cited.  The proof can be modified to prove a slightly weaker result, with 60 replaced by 90 and an assumption that the median of is at least (a constant).  This is remarked in the published version of this chapter, Random walks on quasirandom graphs.

¹ “A note … ” was fairly described by one of my examiners as a “potboiler”.  It was also my first submitted paper, and completed in my second of three years.  Perhaps this will reassure anybody in the first year of a PhD who is worrying that they have yet to publish.

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 -decomposition. In practice, the current obstacle to improving the bounds on is usually our knowledge of another quantity, the fractional decomposition threshold for cliques.

A graph has a fractional -decomposition if we can assign a non-negative weight to each copy of in such that the total weight of the copies of containing each fixed edge of is exactly 1. We prove that every graph with minimum degree at least has a fractional -decomposition. This greatly improves the previous bound of for large . We also prove a similar result for hypergraphs.

The proof begins with an approximate fractional -decomposition obtained by weighting every -clique in our graph equally. We then use small gadgets to make local adjustments to the total weight over each edge until we end up with a genuine fractional -decomposition.

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 call these conditions triangle divisibility. Triangle divisibility is not a sufficient condition for triangle decomposition (consider ), but it is sufficient if is complete. So we would like to know how far from complete can be and triangle divisibility still remain sufficient for triangle decomposition. Nash-Williams conjectured that minimum degree (where is the number of vertices of ) should suffice for large . In this paper we prove that every triangle divisible graph with minimum degree has a triangle decomposition. We also prove similar results with any graph in place of triangles.

The proof uses the absorbing method. It is very easy to remove triangles at the beginning of the process, but very hard at the end. So we make use of the flexibility we have at the beginning to make a plan for dealing with a small remainder. The key idea is that given a possible remainder we can find a graph such that and both have triangle decompositions. By reserving sufficiently many such A at the start of the process we know that we will be able to solve our problems at the end.

In Partition regularity in the rationals we (Barber, Hindman and Leader) showed that there are systems of equations that are partition regular over but not over . Here we show that this separation is very strong: there is an uncountable chain of subgroups from to such that each subgroup has a system that is partition regular over it but over no earlier subgroup. We use our new central sets approach, but this result could also have been proved using the original density method.

Most of the work in this paper is spent proving that the systems we construct are strongly partition regular, in the sense that the variables can be forced to take different values. If you only want to see the application then you can skip part of the argument without losing anything.

Rado’s theorem states that a finite matrix is partition regular if and only if it has the “columns property”. It is easy to write down infinite matrices with the columns property that are not partition regular, but all known examples of partition regular matrices do have the columns property. In this paper we describe a matrix that is partition regular but fails to have the columns property in the strongest possible sense.

The main contribution of this paper is a translation of the key lemma of “Partition regularity in the rationals” to work with central, rather than dense, sets. Central sets have very strong combinatorial properties; for example, they contain solutions to all finite partition regular systems. As a result, our theorems are harder to prove but easier to apply—for the application above we could have proved the partition regularity of the systems using density methods, but the argument would have been more involved.