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Abstract Let P be a poset of size 2 k that has a greatest and a least element. We prove that, for sufficiently large n , the Boolean lattice 2 [ n ] can be partitioned into copies of P. This resolves a conjecture of Lonc. [ABSTRACT FROM AUTHOR]

Abstract: How large can a family be if it does not contain with ? Our aim in this paper is to show that any such family has size at most . This is tight up to a multiplicative constant of 2. We also obtain similar results for families with , showing that they satisfy , where is a constant depending only on . [Copyright &y& Elsevier]

Abstract: Let be a family of subsets of an -set such that does not contain distinct sets and with . How large can be? Our aim in this note is to determine the maximum size of such an . This answers a question of Kalai. We also give some related results and conjectures. [Copyright &y& Elsevier]

Our aim in this note is to 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 Turán problems in the hypercube, but they are also natural in their own right. We start by giving the daisy conjectures, and some related problems, and shall then go on to describe the connection with vertex-Turán problems in the hypercube. [ABSTRACT FROM PUBLISHER]

Abstract: A k-signed r-set on is an ordered pair , where A is an r-subset of and f is a function from A to . Families are said to be cross-intersecting if any set in any family intersects any set in any other family . Hilton proved a sharp bound for the sum of sizes of cross-intersecting families of r-subsets of . Our aim is to generalise Hilton''s bound to one for families of k-signed r-sets on . The main tool developed is an extension of Katona''s cyclic permutation argument. [Copyright &y& Elsevier]

Abstract: Our main aim in this paper is to show that, in Conway''s Angel and Devil game, an Angel of sufficient speed can always escape in three dimensions. We also prove some related results and make some conjectures. [Copyright &y& Elsevier]

It is proved that a connected infinite graph has a normal spanning tree (the infinite analogue of a depth-first search tree) if and only if it has no minor obtained canonically from either an (50, 51)-regular bipartite graph or an order-theoretic Aronszajn tree. This disproves Halin's conjecture that only the first of these obstructions was needed to characterize the graphs with normal spanning trees. As a corollary Halin's further conjecture is deduced, that a connected graph has a normal spanning tree if and only if all its minors have countable colouring number.The precise classification of the (50, 51)-regular bipartite graphs remains an open problem. One such class turns out to contain obvious infinite minor-antichains, so as an unexpected corollary Thomas's result that the infinite graphs are not well-quasi-ordered as minors is reobtained. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]