102 research outputs found

    Partition regularity with congruence conditions

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    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

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    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 22. 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

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    A path in the hypercube QnQ_n is said to be a geodesic if no two of its edges are in the same direction. Let GG be a subgraph of QnQ_n with average degree dd. How long a geodesic must GG contain? We show that GG must contain a geodesic of length dd. 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

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    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 nn vertices) be? We show that there exist oriented 3-graphs whose shortest cycle has length n22(1+o(1))\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 n23(1+o(1))\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

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    For a fixed rr, let fr(n)f_r(n) denote the minimum number of complete rr-partite rr-graphs needed to partition the complete rr-graph on nn vertices. The Graham-Pollak theorem asserts that f2(n)=n1f_2(n)=n-1. An easy construction shows that fr(n)(1+o(1))(nr/2)f_r(n) \leq (1+o(1))\binom{n}{\lfloor r/2 \rfloor}, and we write crc_r for the least number such that fr(n)cr(1+o(1))(nr/2)f_r(n) \leq c_r (1+o(1))\binom{n}{\lfloor r/2 \rfloor}. It was known that cr<1c_r < 1 for each even r4r \geq 4, but this was not known for any odd value of rr. In this short note, we prove that c295<1c_{295}<1. Our method also shows that cr0c_r \rightarrow 0, answering another open problem

    Daisies and Other Turan Problems

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    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

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    It is shown that the maximum number of patterns that can occur in a permutation of length nn is asymptotically 2n2^n. This significantly improves a previous result of Coleman
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