arXiv subjects

Paul A. Russell

Publications and source records attributed to Paul A. Russell.

12 recordsLinked to original sources

arXiv2026

A parallel wakeup problem and multi-room light switch strategies

The wakeup problem in distributed computing asks for a symmetric protocol that enables one of several processors to eventually guarantee that all (or, in a more general setting, enough) other processors have acted, using a shared register but no global clock. Dropping the symmetry requirement gives a well-known exercise often phrased in terms of prisoners entering, in an unknown sequence, a room equipped with a single binary switch, and using it to communicate. Kane and Kominers recently analysed a more general version of the latter with multiple parallel and indistinguishable rooms. We answer some open questions of Kane and Kominers regarding the minimum number of switch states needed for the prisoners to solve the problem. We also consider the symmetric ``wakeup'' version of this scenario, and establish exactly for which numbers of processors and registers a solution is possible.

cs.DC↗

arXiv2019

Infinite monochromatic sumsets for colourings of the reals

N. Hindman, I. Leader and D. Strauss proved that it is consistent that there is a finite colouring of $\mathbb R$ so that no infinite sumset $X+X=\{x+y:x,y\in X\}$ is monochromatic. Our aim in this paper is to prove a consistency result in the opposite direction: we show that, under certain set-theoretic assumptions, for any $c:\mathbb R\to r$ with $r$ finite there is an infinite $X\subseteq \mathbb R$ so that $c$ is constant on $X+X$.

math.LO↗

arXiv2018

Inhomogeneous Partition Regularity

We say that the system of equations $Ax=b$, where $A$ is an integer matrix and $b$ is a (non-zero) integer vector, is partition regular if whenever the integers are finitely coloured there is a monochromatic vector $x$ with $Ax=b$. Rado proved that the system $Ax=b$ is partition regular if and only if it has a constant solution. Byszewski and Krawczyk asked if this remains true when the integers are replaced by a general ring $R$. Our aim in this note is to answer this question in the affirmative. The main ingredient is a new `direct' proof of Rado's result.

math.CO↗

arXiv2017

Monochromatic infinite sumsets

We show that there is a rational vector space $V$ such that, whenever $V$ is finitely coloured, there is an infinite set $X$ whose sumset $X+X$ is monochromatic. Our example is the rational vector space of dimension $\sup\{\aleph_0,2^{\aleph_0},2^{2^{\aleph_0}},\ldots\,\}$. This complements a result of Hindman, Leader and Strauss, who showed that the result does not hold for dimension below $\aleph_ω$. So our result is best possible under GCH.

math.CO↗

arXiv2013

Set Systems Containing Many Maximal Chains

The purpose of this short problem paper is to raise an extremal question on set systems which seems to be natural and appealing. Our question is: which set systems of a given size maximise the number of $(n+1)$-element chains in the power set $\mathcal{P}(\{1,2,\dots,n\})$? We will show that for each fixed $α>0$ there is a family of $α2^n$ sets containing $(α+o(1))n!$ such chains, and that this is asymptotically best possible. For smaller set systems we are unable to answer the question. We conjecture that a `tower of cubes' construction is extremal. We finish by mentioning briefly a connection to an extremal problem on posets and a variant of our question for the grid graph.

math.CO↗

arXiv2011

Probably Intersecting Families are Not Nested

It is well known that an intersecting family of subsets of an n-element set can contain at most 2^(n-1) sets. It is natural to wonder how `close' to intersecting a family of size greater than 2^(n-1) can be. Katona, Katona and Katona introduced the idea of a `most probably intersecting family.' Suppose that X is a family and that 0<p<1. Let X(p) be the (random) family formed by selecting each set in X independently with probability p. A family X is `most probably intersecting' if it maximises the probability that X(p) is intersecting over all families of size |X|. Katona, Katona and Katona conjectured that there is a nested sequence consisting of most probably intersecting families of every possible size. We show that this conjecture is false for every value of p provided that n is sufficiently large.

math.CO↗

arXiv2011

Compressions and Probably Intersecting Families

A family X of sets is said to be intersecting if any two members of X have non-empty intersection. It is a well-known and simple fact that an intersecting family of subsets of [n]={1,2,...,n} can contain at most 2^(n-1) sets. Katona, Katona and Katona ask the following question. Suppose instead a family X of subsets of [n] satisfies |X|=2^(n-1)+i for some fixed i>0. Create a new family X_p by choosing each member of X independently with some fixed probability p. How do we choose X to maximize the probability that X_p is intersecting? They conjecture that there is a nested sequence of optimal families for i=1, 2, ..., 2^(n-1). In this paper, we show that the families [n]^(\ge r)={A\subset[n]:|A|\ge r} are optimal for the appropriate values of i, thereby proving the conjecture for this sequence of values. Moreover, we show that for intermediate values of i there exist optimal families lying between those we have found. It turns out that the optimal families we find simultaneously maximize the number of intersecting subfamilies of every possible order. Standard compression techniques appear inadequate to solve the problem as they do not preserve intersection properties of subfamilies. Instead, our main tool is a novel compression method, together with a way of `compressing' subfamilies, which may be of independent interest.

math.CO↗

arXiv2010

Transitive Sets and Cyclic Quadrilaterals

Motivated by some questions in Euclidean Ramsey theory, our aim in this note is to show that there exists a cyclic quadrilateral that does not embed into any transitive set (in any dimension). We show that in fact this holds for almost all cyclic quadrilaterals, and we also give explicit examples of such cyclic quadrilaterals. These are the first explicit examples of spherical sets that do not embed into transitive sets.

math.CO↗

arXiv2010

Transitive Sets in Euclidean Ramsey Theory

A finite set $X$ in some Euclidean space $R^n$ is called Ramsey if for any $k$ there is a $d$ such that whenever $R^d$ is $k$-coloured it contains a monochromatic set congruent to $X$. This notion was introduced by Erdos, Graham, Montgomery, Rothschild, Spencer and Straus, who asked if a set is Ramsey if and only if it is spherical, meaning that it lies on the surface of a sphere. This question (made into a conjecture by Graham) has dominated subsequent work in Euclidean Ramsey theory. In this paper we introduce a new conjecture regarding which sets are Ramsey; this is the first ever `rival' conjecture to the conjecture above. Calling a finite set transitive if its symmetry group acts transitively---in other words, if all points of the set look the same---our conjecture is that the Ramsey sets are precisely the transitive sets, together with their subsets. One appealing feature of this conjecture is that it reduces (in one direction) to a purely combinatorial statement. We give this statement as well as several other related conjectures. We also prove the first non-trivial cases of the statement. Curiously, it is far from obvious that our new conjecture is genuinely different from the old. We show that they are indeed different by proving that not every spherical set embeds in a transitive set. This result may be of independent interest.

math.CO↗

arXiv2007

Families intersecting on an interval

We shall be interested in the following Erdos-Ko-Rado-type question. Fix some subset B of [n]. How large a family A of subsets of [n] can we find such that the intersection of any two sets in A contains a cyclic translate (modulo n) of B? Chung, Graham, Frankl and Shearer have proved that, in the case where B is a block of length t, we can do no better than to take A to consist of all supersets of B. We give an alternative proof of this result, which is in a certain sense more 'direct'.

math.CO↗

arXiv2007

Bayesian modelling of the cool core galaxy group NGC 4325

We present an X-ray analysis of the radio-quiet cool-core galaxy group NGC 4325 (z=0.026) based on Chandra and ROSAT observations. The Chandra data were analysed using XSPEC deprojection, 2D spectral mapping and forward-fitting with parametric models. Additionally, a Markov chain Monte Carlo method was used to perform a joint Bayesian analysis of the Chandra and ROSAT data. The results of the various analysis methods are compared, particularly those obtained by forward-fitting and deprojection. The spectral mapping reveals the presence of cool gas displaced up to 10 kpc from the group centre. The Chandra X-ray surface brightness shows the group core to be highly disturbed, and indicates the presence of two small X-ray cavities within 15 kpc of the group core. The XSPEC deprojection analysis shows that the group has a particularly steep entropy profile, suggesting that an AGN outburst may be about to occur. With the evidence of prior AGN activity, but with no radio emission currently observed, we suggest that the group in in a pre-outburst state, with the cavities and displaced gas providing evidence of a previous, weak AGN outburst.

astro-ph↗

arXiv2006

The Group Evolution Multiwavelength Study (GEMS): The near-infrared luminosity function of nearby galaxy groups

We present J and K-band luminosity functions (LF) for the Group Evolution Multiwavelength Study (GEMS) sample of 60 nearby groups of galaxies, with photometry from the 2MASS survey. We find that, as seen in B and R-band photometry of a subsample of these groups in our earlier work, the LFs of the X-ray dim groups (L_X < 10^41.7 erg/s) show a depletion of galaxies of intermediate luminosity around M_K = -23, within a radius 0.3R_500 from the centres of these groups. This feature is not seen in the X-ray brighter groups, nor in either kind of group when the LFs are determined all the way out to R_500. We conclude that an enhanced level of star formation is not responsible for the this feature. From the faint end of the LFs, we find support for the under-abundance of low surface brightness dwarfs in the 2MASS survey. We find that for all kinds of groups, the modelling of the luminosity function, with universal forms for the LFs of galaxies of different morphological types, fails when simultaneously required to fit the B and K-band LFs. This means that the dip-like features seen in LFs are not merely due to the varying proportions of galaxies of different morphological types among the X-ray dim and bright groups. We argue that this support our hypothesis that this feature is due to the enhanced merging of intermediate-mass galaxies in the dynamically sluggish environment of X-ray dim groups.

astro-ph↗