Induced Ramsey-type results and binary predicates for point sets

Let $k$ and $p$ be positive integers and let $Q$ be a finite point set in general position in the plane. We say that $Q$ is $(k,p)$-Ramsey if there is a finite point set $P$ such that for every $k$-coloring $c$ of $\binom{P}{p}$ there is a subset $Q'$ of $P$ such that $Q'$ and $Q$ have the same order type and $\binom{Q'}{p}$ is monochromatic in $c$. Ne\v{s}et\v{r}il and Valtr proved that for every $k \in \mathbb{N}$, all point sets are $(k,1)$-Ramsey. They also proved that for every $k \ge 2$ and $p \ge 2$, there are point sets that are not $(k,p)$-Ramsey. As our main result, we introduce a new family of $(k,2)$-Ramsey point sets, extending a result of Ne\v{s}et\v{r}il and Valtr. We then use this new result to show that for every $k$ there is a point set $P$ such that no function $\Gamma$ that maps ordered pairs of distinct points from $P$ to a set of size $k$ can satisfy the following "local consistency" property: if $\Gamma$ attains the same values on two ordered triples of points from $P$, then these triples have the same orientation. Intuitively, this implies that there cannot be such a function that is defined locally and determines the orientation of point triples.Comment: 22 pages, 3 figures, final version, minor correction

Binary black hole mergers from field triples: properties, rates and the impact of stellar evolution

We consider the formation of binary black hole mergers through the evolution of field massive triple stars. In this scenario, favorable conditions for the inspiral of a black hole binary are initiated by its gravitational interaction with a distant companion, rather than by a common-envelope phase invoked in standard binary evolution models. We use a code that follows self-consistently the evolution of massive triple stars, combining the secular triple dynamics (Lidov-Kozai cycles) with stellar evolution. After a black hole triple is formed, its dynamical evolution is computed using either the orbit-averaged equations of motion, or a high-precision direct integrator for triples with weaker hierarchies for which the secular perturbation theory breaks down. Most black hole mergers in our models are produced in the latter non-secular dynamical regime. We derive the properties of the merging binaries and compute a black hole merger rate in the range (0.3- 1.3) Gpc^{-3}yr^{-1}, or up to ~2.5Gpc^{-3}yr^{-1} if the black hole orbital planes have initially random orientation. Finally, we show that black hole mergers from the triple channel have significantly higher eccentricities than those formed through the evolution of massive binaries or in dense star clusters. Measured eccentricities could therefore be used to uniquely identify binary mergers formed through the evolution of triple stars. While our results suggest up to ~10 detections per year with Advanced-LIGO, the high eccentricities could render the merging binaries harder to detect with planned space based interferometers such as LISA.Comment: Accepted for publication in ApJ. 10 pages, 6 figure

Predictions for Triple Stars with and without a Pulsar in Star Clusters

Though about 80 pulsar binaries have been detected in globular clusters so far, no pulsar has been found in a triple system in which all three objects are of comparable mass. Here we present predictions for the abundance of such triple systems, and for the most likely characteristics of these systems. Our predictions are based on an extensive set of more than 500 direct simulations of star clusters with primordial binaries, and a number of additional runs containing primordial triples. Our simulations employ a number N_{tot} of equal mass stars from N_{tot}=512 to N_{tot}=19661 and a primordial binary fraction from 0-50%. In addition, we validate our results against simulations with N=19661 that include a mass spectrum with a turn-off mass at 0.8 M_{sun}, appropriate to describe the old stellar populations of galactic globular clusters. Based on our simulations, we expect that typical triple abundances in the core of a dense cluster are two orders of magnitude lower than the binary abundances, which in itself already suggests that we don't have to wait too long for the first comparable-mass triple with a pulsar to be detected.Comment: 11 pages, minor changes to match MNRAS accepted versio

Majority voting on restricted domains

In judgment aggregation, unlike preference aggregation, not much is known about domain restrictions that guarantee consistent majority outcomes. We introduce several conditions on individual judgments sufficient for consistent majority judgments. Some are based on global orders of propositions or individuals, others on local orders, still others not on orders at all. Some generalize classic social-choice-theoretic domain conditions, others have no counterpart. Our most general condition generalizes Sen's triplewise value-restriction, itself the most general classic condition. We also prove a new characterization theorem: for a large class of domains, if there exists any aggregation function satisfying some democratic conditions, then majority voting is the unique such function. Taken together, our results support the robustness of majority rule

Algebraic foundations for qualitative calculi and networks

A qualitative representation $\phi$ is like an ordinary representation of a relation algebra, but instead of requiring $(a; b)^\phi = a^\phi | b^\phi$, as we do for ordinary representations, we only require that $c^\phi\supseteq a^\phi | b^\phi \iff c\geq a ; b$, for each $c$ in the algebra. A constraint network is qualitatively satisfiable if its nodes can be mapped to elements of a qualitative representation, preserving the constraints. If a constraint network is satisfiable then it is clearly qualitatively satisfiable, but the converse can fail. However, for a wide range of relation algebras including the point algebra, the Allen Interval Algebra, RCC8 and many others, a network is satisfiable if and only if it is qualitatively satisfiable. Unlike ordinary composition, the weak composition arising from qualitative representations need not be associative, so we can generalise by considering network satisfaction problems over non-associative algebras. We prove that computationally, qualitative representations have many advantages over ordinary representations: whereas many finite relation algebras have only infinite representations, every finite qualitatively representable algebra has a finite qualitative representation; the representability problem for (the atom structures of) finite non-associative algebras is NP-complete; the network satisfaction problem over a finite qualitatively representable algebra is always in NP; the validity of equations over qualitative representations is co-NP-complete. On the other hand we prove that there is no finite axiomatisation of the class of qualitatively representable algebras.Comment: 22 page