# Clique decompositions of multipartite graphs and completion of Latin squares

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.

# 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 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

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.

# Fractional clique decompositions of dense graphs and hypergraphs

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.

# Edge-decompositions of graphs with high minimum degree

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.

# Distinguishing subgroups of the rationals by their Ramsey properties

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.

# Partition regularity without the columns property

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.

# Partition regularity of a system of De and Hindman

De and Hindman proposed that a particular system should be partition regular but not partition regular near zero. With Neil Hindman and Imre Leader I found a different example; in this paper I show that De and Hindman’s original system also works.

# Partition regularity with congruence conditions

Does a partition regular system remain partition regular if we ask that each variable is divisible by ? Not necessarily. This answers several open questions from Hindman, Leader and Strauss’s 2003 survey.

The proof of Proposition 5 in the journal version is not entirely clear; I recommend reading the updated pdf linked above. My thanks to Boaz Tsaban for pointing this out.

# Partition regularity in the rationals

A system of linear equations is partition regular if, whenever the natural numbers are finitely coloured, the system of equations has a monochromatic solution. Partition regularity can also be defined over the rationals, and if the system of equations is finite then these notions coincide. We construct an example of an infinite system which is partition regular over the rationals but not the naturals. The proof is based on examining what happens when you take iterated sumsets and difference sets of subsets of the integers with positive upper density.

# Random walks on quasirandom graphs

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 pick up the same number of random edges at every vertex. In the case where the minimum degree of is large, this argument is essentially correct. If has some vertices of very low degree then it breaks down because the random walk can get stuck in clusters of low degree vertices. However, a more sophisticated argument can recover a result that is almost as strong.

The proofs both fall into two parts: first show that the random walk does not differ too much from a process that has much more independence, then exploit that independence by applying standard concentration results to show that things work with high probability. It turns out that our results can be tweaked to apply to the more general case of random homomorphisms of trees (rather than paths) provided the maximum degree of the tree isn’t too large, so we indicate the necessary changes at the end of the paper.