Matchings and Hamilton Cycles with Constraints on Sets of Edges

The aim of this paper is to extend and generalise some work of Katona on the existence of perfect matchings or Hamilton cycles in graphs subject to certain constraints. The most general form of these constraints is that we are given a family of sets of edges of our graph and are not allowed to use all the edges of any member of this family. We consider two natural ways of expressing constraints of this kind using graphs and using set systems. For the first version we ask for conditions on regular bipartite graphs $G$ and $H$ for there to exist a perfect matching in $G$, no two edges of which form a $4$-cycle with two edges of $H$. In the second, we ask for conditions under which a Hamilton cycle in the complete graph (or equivalently a cyclic permutation) exists, with the property that it has no collection of intervals of prescribed lengths whose union is an element of a given family of sets. For instance we prove that the smallest family of $4$-sets with the property that every cyclic permutation of an $n$-set contains two adjacent pairs of points has size between $(1/9+o(1))n^2$ and $(1/2-o(1))n^2$. We also give bounds on the general version of this problem and on other natural special cases. We finish by raising numerous open problems and directions for further study.Comment: 21 page

Tur\'an and Ramsey Properties of Subcube Intersection Graphs

The discrete cube $\{0,1\}^d$ is a fundamental combinatorial structure. A subcube of $\{0,1\}^d$ is a subset of $2^k$ of its points formed by fixing $k$ coordinates and allowing the remaining $d-k$ to vary freely. The subcube structure of the discrete cube is surprisingly complicated and there are many open questions relating to it. This paper is concerned with patterns of intersections among subcubes of the discrete cube. Two sample questions along these lines are as follows: given a family of subcubes in which no $r+1$ of them have non-empty intersection, how many pairwise intersections can we have? How many subcubes can we have if among them there are no $k$ which have non-empty intersection and no $l$ which are pairwise disjoint? These questions are naturally expressed as Tur\'an and Ramsey type questions in intersection graphs of subcubes where the intersection graph of a family of sets has one vertex for each set in the family with two vertices being adjacent if the corresponding subsets intersect. Tur\'an and Ramsey type problems are at the heart of extremal combinatorics and so these problems are mathematically natural. However, a second motivation is a connection with some questions in social choice theory arising from a simple model of agreement in a society. Specifically, if we have to make a binary choice on each of $n$ separate issues then it is reasonable to assume that the set of choices which are acceptable to an individual will be represented by a subcube. Consequently, the pattern of intersections within a family of subcubes will have implications for the level of agreement within a society. We pose a number of questions and conjectures relating directly to the Tur\'an and Ramsey problems as well as raising some further directions for study of subcube intersection graphs.Comment: 18 page

Mixing fuel particles for space combustion research using acoustics

Part of the microgravity science to be conducted aboard the Shuttle (STS) involves combustion using solids, particles, and liquid droplets. The central experimental facts needed for characterization of premixed quiescent particle cloud flames cannot be adequately established by normal gravity studies alone. The experimental results to date of acoustically mixing a prototypical particulate, lycopodium, in a 5 cm diameter by 75 cm long flame tube aboard a Learjet aircraft flying a 20 sec low gravity trajectory are described. Photographic and light detector instrumentation combine to measure and characterize particle cloud uniformity

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 $\alpha>0$ there is a family of $\alpha 2^n$ sets containing $(\alpha+o(1))n!$ 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.Comment: 5 page

Disparities in Cause-Specific Cancer Survival by Census Tract Poverty Level in Idaho, U.S.

Objective. This population-based study compared cause-specific cancer survival by socioeconomic status using methods to more accurately assign cancer deaths to primary site. Methods. The current study analyzed Idaho data used in the Accuracy of Cancer Mortality Statistics Based on Death Certificates (ACM) study supplemented with additional information to measure cause-specific cancer survival by census tract poverty level. Results. The distribution of cases by primary site group differed significantly by poverty level (chi-square = 265.3, 100 df, p In the life table analyses, for 8 of 24 primary site groups investigated, and all sites combined, there was a significant gradient relating higher poverty with poorer survival. For all sites combined, the absolute difference in 5-year cause-specific survival rate was 13.6% between the lowest and highest poverty levels. Conclusions. This study shows striking disparities in cause-specific cancer survival related to the poverty level of the area a person resides in at the time of diagnosis