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Given a countable dense subset S of a finite-dimensional normed space X, and 0 < p < 1, we form a random graph on S by joining, independently and with probability p, each pair of points at distance less than 1. We say that S is Rado if any two such random graphs are (almost surely) isomorphic. Bonato and Janssen showed that in ℓ∞d almost all S are Rado. Our main aim in this paper is to show that ℓ∞d is the unique normed space with this property: indeed, in every other space almost all sets S are non-Rado. We also determine which spaces admit some Rado set: this turns out to be the spaces that have an ℓ∞ direct summand. These results answer questions of Bonato and Janssen. A key role is played by the determination of which finite-dimensional normed spaces have the property that every bijective step-isometry (meaning that the integer part of distances is preserved) is in fact an isometry. This result may be of independent interest. [ABSTRACT FROM AUTHOR]

A finite or infinite matrix A with rational entries is called partition regular if whenever the natural numbers are finitely coloured there is a monochromatic vector x with Ax = 0. Many of the classical theorems of Ramsey 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. [ABSTRACT FROM AUTHOR]