Induced restricted Ramsey theorems for spaces

AbstractThe induced restricted versions of the vector space Ramsey theorem and of the Graham-Rothschild parameter set theorem are proved

Fluorescent non-toxic bait as a new method for black rat (Rattus rattus) monitoring

The detection of synathropic rodents may be difficult since they are animals with nocturnal activity. Methods of their detection and monitoring rely mostly on indirect signs of their activity such as the presence of faeces, urine, consumed foods and damaged materials. Our experimental hypothesis was that the production of fluorescent faeces - following consumption of fluorescent bait - may be used for rodent monitoring. For this purpose we studied the production of fluorescent faeces, temporal dynamics and detectability in wild black rat (Rattus rattus). Wild black rats were individually housed in experimental cages with the wire-mesh grid floor and faeces were collected in short-time intervals. The peak of fluorescent activity in faeces was detected 10-20 hours after bait ingestion. We found that there is only relatively short delay between bait consumption and defecation and fluorescent faeces are easily detectable at distance using an ultraviolet hand lamp. Thus, this method can contribute to effective monitoring of rodent pests.Keywords: Rattus rattus, Fluorescent bait, Monitoring, Rodent contro

Near-optimum universal graphs for graphs with bounded degrees (Extended abstract)

Let H be a family of graphs. We say that G is H-universal if, for each H ∈H, the graph G contains a subgraph isomorphic to H. Let H(k, n) denote the family of graphs on n vertices with maximum degree at most k. For each fixed k and each n sufficiently large, we explicitly construct an H(k, n)-universal graph Γ(k, n) with O(n2−2/k(log n)1+8/k) edges. This is optimal up to a small polylogarithmic factor, as Ω(n2−2/k) is a lower bound for the number of edges in any such graph. En route, we use the probabilistic method in a rather unusual way. After presenting a deterministic construction of the graph Γ(k, n), we prove, using a probabilistic argument, that Γ(k, n) is H(k, n)-universal. So we use the probabilistic method to prove that an explicit construction satisfies certain properties, rather than showing the existence of a construction that satisfies these properties. © Springer-Verlag Berlin Heidelberg 200

A separator theorem for hypergraphs and a CSP-SAT algorithm

We show that for every r≥2 there exists ϵr>0 such that any r-uniform hypergraph with m edges and maximum vertex degree o(m−−√) contains a set of at most (12−ϵr)m edges the removal of which breaks the hypergraph into connected components with at most m/2 edges. We use this to give an algorithm running in time d(1−ϵr)m that decides satisfiability of m-variable (d,k)-CSPs in which every variable appears in at most r constraints, where ϵr depends only on r and k∈o(m−−√). Furthermore our algorithm solves the corresponding #CSP-SAT and Max-CSP-SAT of these CSPs. We also show that CNF representations of unsatisfiable (2,k)-CSPs with variable frequency r can be refuted in tree-like resolution in size 2(1−ϵr)m. Furthermore for Tseitin formulas on graphs with degree at most k (which are (2,k)-CSPs) we give a deterministic algorithm finding such a refutation

Testing Linear-Invariant Non-Linear Properties

We consider the task of testing properties of Boolean functions that are invariant under linear transformations of the Boolean cube. Previous work in property testing, including the linearity test and the test for Reed-Muller codes, has mostly focused on such tasks for linear properties. The one exception is a test due to Green for "triangle freeness": a function f:\cube^{n}\to\cube satisfies this property if $f(x),f(y),f(x+y)$ do not all equal 1, for any pair x,y\in\cube^{n}. Here we extend this test to a more systematic study of testing for linear-invariant non-linear properties. We consider properties that are described by a single forbidden pattern (and its linear transformations), i.e., a property is given by $k$ points v_{1},...,v_{k}\in\cube^{k} and f:\cube^{n}\to\cube satisfies the property that if for all linear maps L:\cube^{k}\to\cube^{n} it is the case that $f(L(v_{1})),...,f(L(v_{k}))$ do not all equal 1. We show that this property is testable if the underlying matroid specified by $v_{1},...,v_{k}$ is a graphic matroid. This extends Green's result to an infinite class of new properties. Our techniques extend those of Green and in particular we establish a link between the notion of "1-complexity linear systems" of Green and Tao, and graphic matroids, to derive the results.Comment: This is the full version; conference version appeared in the proceedings of STACS 200

Upper tails for counting objects in randomly induced subhypergraphs and rooted random graphs

General upper tail estimates are given for counting edges in a random induced subhypergraph of a fixed hypergraph H, with an easy proof by estimating the moments. As an application we consider the numbers of arithmetic progressions and Schur triples in random subsets of integers. In the second part of the paper we return to the subgraph counts in random graphs and provide upper tail estimates in the rooted case.Comment: 15 page

Quasirandom permutations are characterized by 4-point densities

For permutations π and τ of lengths |π|≤|τ| , let t(π,τ) be the probability that the restriction of τ to a random |π| -point set is (order) isomorphic to π . We show that every sequence {τj} of permutations such that |τj|→∞ and t(π,τj)→1/4! for every 4-point permutation π is quasirandom (that is, t(π,τj)→1/|π|! for every π ). This answers a question posed by Graham