Ben Barber

Fractional Clique Decompositions of Dense Graphs and Hypergraphs

Abstract Our main result is that every graph G on n ≥ 10 4 r 3 vertices with minimum degree δ ( G ) ≥ ( 1 − 1 / 10 4 r 3 / 2 ) n has a fractional K r -decomposition. Combining this result with recent work of Barber, Kuhn, Lo and Osthus leads to the best known minimum degree thresholds for exact (non-fractional) F-decompositions for a wide class of graphs F (including large cliques). For general k-uniform hypergraphs, we give a short argument which shows that there exists a constant c k > 0 such that every k-uniform hypergraph G on n vertices with minimum codegree at least ( 1 − c k / r 2 k − 1 ) n has a fractional K r ( k ) -decomposition, where K r ( k ) is the complete k-uniform hypergraph on r vertices. (Related fractional decomposition results for triangles have been obtained by Dross and for hypergraph cliques by Dukes as well as Yuster.) All the above new results involve purely combinatorial arguments. In particular, this yields a combinatorial proof of Wilsons theorem that every large F-divisible complete graph has an F-decomposition.

Clique decompositions of multipartite graphs and completion of latin squares

Our main result essentially reduces the problem of finding an edge-decomposition of a balanced r-partite graph of large minimum degree into r-cliques to the problem of finding a fractional r-clique decomposition or an approximate one. Together with very recent results of Bowditch and Dukes as well as Montgomery on fractional decompositions into triangles and cliques respectively, this gives the best known bounds on the minimum degree which ensures an edge-decomposition of an r-partite graph into r-cliques (subject to trivially necessary divisibility conditions). The case of triangles translates into the setting of partially completed Latin squares and more generally the case of r-cliques translates into the setting of partially completed mutually orthogonal Latin squares.

Partition regularity without the columns property

A finite or infinite matrix A with rational entries is called parti- tion regular if whenever the natural numbers are finitely coloured there is a monochromatic vector x with Ax = 0. Many of the classical theorems of Ram- sey Theory may naturally be interpreted as assertions that particular matrices are partition regular. In the finite case, Rado proved that a matrix is partition regular if and only it satisfies a computable condition known as the columns property. The first requirement of the columns property is that some set of columns sums to zero. In the infinite case, much less is known. There are many examples of matrices with the columns property that are not partition regular, but until now all known examples of partition regular matrices did have the columns property. Our main aim in this paper is to show that, perhaps surprisingly, there are infinite partition regular matrices without the columns property — in fact, having no set of columns summing to zero. We also make a conjecture that if a partition regular matrix (say with integer coefficients) has bounded row sums then it must have the columns property, and prove a first step towards this.

Partition regularity in the rationals

Abstract A system of homogeneous linear equations with integer coefficients is partition regular if, whenever the natural numbers are finitely coloured, the system has a monochromatic solution. The Finite Sums theorem provided the first example of an infinite partition regular system of equations. Since then, other such systems of equations have been found, but each can be viewed as a modification of the Finite Sums theorem. We present here a new infinite partition regular system of equations that appears to arise in a genuinely different way. This is the first example of a partition regular system in which a variable occurs with unbounded coefficients. A modification of the system provides an example of a system that is partition regular over Q but not N , settling another open problem.

Distinguishing subgroups of the rationals by their Ramsey properties

A system of linear equations with integer coefficients is partition regular over a subset S of the reals if, whenever S ? { 0 } is finitely coloured, there is a solution to the system contained in one colour class. It has been known for some time that there is an infinite system of linear equations that is partition regular over R but not over Q , and it was recently shown (answering a long-standing open question) that one can also distinguish Q from Z in this way.Our aim is to show that the transition from Z to Q is not sharp: there is an infinite chain of subgroups of Q , each of which has a system that is partition regular over it but not over its predecessors. We actually prove something stronger: our main result is that if R and S are subrings of Q with R not contained in S, then there is a system that is partition regular over R but not over S. This implies, for example, that the chain above may be taken to be uncountable.

Maximum Hitting for n Sufficiently Large

For a left-compressed intersecting family

Edge-decompositions of graphs with high minimum degree

{\fancyscript{A} \subseteq[n]^{(r)}}

Minimalist designs.

and a set