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Abstract: We show that any subgraph of the hypercube of average degree contains a geodesic of length , where by geodesic we mean a shortest path in . This result, which is best possible, strengthens a theorem of Feder and Subi. It is also related to the ‘antipodal colourings’ conjecture of Norine. [Copyright &y& Elsevier]

Abstract: Let be a finite matrix with rational entries. We say that is doubly image partition regular if whenever the set of positive integers is finitely coloured, there exists such that the entries of are all the same colour (or monochromatic) and also, the entries of are monochromatic. Which matrices are doubly image partition regular? More generally, we say that a pair of matrices , where and have the same number of rows, is doubly kernel partition regular if whenever is finitely coloured, there exist vectors and , each monochromatic, such that . (So the case above is the case when is the negative of the identity matrix.) There is an obvious sufficient condition for the pair to be doubly kernel partition regular, namely that there exists a positive rational such that the matrix is kernel partition regular. (That is, whenever is finitely coloured, there exists monochromatic such that .) Our aim in this paper is to show that this sufficient condition is also necessary. As a consequence we have that a matrix is doubly image partition regular if and only if there is a positive rational such that the matrix is kernel partition regular, where is the identity matrix of the appropriate size. We also prove extensions to the case of several matrices. [Copyright &y& Elsevier]