Partition regularity with congruence conditions

An infinite integer matrix A is called image partition regular if, whenever the natural numbers are finitely coloured, there is an integer vector x such that Ax is monochromatic. Given an image partition regular matrix A, can we also insist that each variable x_i is a multiple of some given d_i? This is a question of Hindman, Leader and Strauss. Our aim in this short note is to show that the answer is negative. As an application, we disprove a conjectured equivalence between the two main forms of partition regularity, namely image partition regularity and kernel partition regularity.Comment: 5 page

Forbidding a Set Difference of Size 1

How large can a family \cal A \subset \cal P [n] be if it does not contain A,B with |A\setminus B| = 1? Our aim in this paper is to show that any such family has size at most \frac{2+o(1)}{n} \binom {n}{\lfloor n/2\rfloor }. This is tight up to a multiplicative constant of $2$. We also obtain similar results for families \cal A \subset \cal P[n] with |A\setminus B| \neq k, showing that they satisfy |{\mathcal A}| \leq \frac{C_k}{n^k}\binom {n}{\lfloor n/2\rfloor }, where C_k is a constant depending only on k.Comment: 8 pages. Extended to include bound for families \cal A \subset \cal P [n] satisfying |A\setminus B| \neq k for all A,B \in \cal

Long geodesics in subgraphs of the cube

A path in the hypercube $Q_n$ is said to be a geodesic if no two of its edges are in the same direction. Let $G$ be a subgraph of $Q_n$ with average degree $d$. How long a geodesic must $G$ contain? We show that $G$ must contain a geodesic of length $d$. This result, which is best possible, strengthens a theorem of Feder and Subi. It is also related to the `antipodal colourings' conjecture of Norine.Comment: 8 page

Cycles in Oriented 3-graphs

An oriented 3-graph consists of a family of triples (3-sets), each of which is given one of its two possible cyclic orientations. A cycle in an oriented 3-graph is a positive sum of some of the triples that gives weight zero to each 2-set. Our aim in this paper is to consider the following question: how large can the girth of an oriented 3-graph (on $n$ vertices) be? We show that there exist oriented 3-graphs whose shortest cycle has length $\frac{n^2}{2}(1+o(1))$: this is asymptotically best possible. We also show that there exist 3-tournaments whose shortest cycle has length $\frac{n^2}{3}(1+o(1))$, in complete contrast to the case of 2-tournaments.Comment: 12 page

Improved Bounds for the Graham-Pollak Problem for Hypergraphs

For a fixed $r$, let $f_r(n)$ denote the minimum number of complete $r$-partite $r$-graphs needed to partition the complete $r$-graph on $n$ vertices. The Graham-Pollak theorem asserts that $f_2(n)=n-1$. An easy construction shows that $f_r(n) \leq (1+o(1))\binom{n}{\lfloor r/2 \rfloor}$, and we write $c_r$ for the least number such that $f_r(n) \leq c_r (1+o(1))\binom{n}{\lfloor r/2 \rfloor}$. It was known that $c_r < 1$ for each even $r \geq 4$, but this was not known for any odd value of $r$. In this short note, we prove that $c_{295}<1$. Our method also shows that $c_r \rightarrow 0$, answering another open problem

Daisies and Other Turan Problems

We make some conjectures about extremal densities of daisy-free families, where a `daisy' is a certain hypergraph. These questions turn out to be related to some Turan problems in the hypercube, but they are also natural in their own right

Permutations Containing Many Patterns

It is shown that the maximum number of patterns that can occur in a permutation of length $n$ is asymptotically $2^n$. This significantly improves a previous result of Coleman