Vertex Ramsey problems in the hypercube

If we 2-color the vertices of a large hypercube what monochromatic substructures are we guaranteed to find? Call a set S of vertices from Q_d, the d-dimensional hypercube, Ramsey if any 2-coloring of the vertices of Q_n, for n sufficiently large, contains a monochromatic copy of S. Ramsey's theorem tells us that for any r \geq 1 every 2-coloring of a sufficiently large r-uniform hypergraph will contain a large monochromatic clique (a complete subhypergraph): hence any set of vertices from Q_d that all have the same weight is Ramsey. A natural question to ask is: which sets S corresponding to unions of cliques of different weights from Q_d are Ramsey? The answer to this question depends on the number of cliques involved. In particular we determine which unions of 2 or 3 cliques are Ramsey and then show, using a probabilistic argument, that any non-trivial union of 39 or more cliques of different weights cannot be Ramsey. A key tool is a lemma which reduces questions concerning monochromatic configurations in the hypercube to questions about monochromatic translates of sets of integers.Comment: 26 pages, 3 figure

Polychromatic Colorings on the Hypercube

Given a subgraph G of the hypercube Q_n, a coloring of the edges of Q_n such that every embedding of G contains an edge of every color is called a G-polychromatic coloring. The maximum number of colors with which it is possible to G-polychromatically color the edges of any hypercube is called the polychromatic number of G. To determine polychromatic numbers, it is only necessary to consider a structured class of colorings, which we call simple. The main tool for finding upper bounds on polychromatic numbers is to translate the question of polychromatically coloring the hypercube so every embedding of a graph G contains every color into a question of coloring the 2-dimensional grid so that every so-called shape sequence corresponding to G contains every color. After surveying the tools for finding polychromatic numbers, we apply these techniques to find polychromatic numbers of a class of graphs called punctured hypercubes. We also consider the problem of finding polychromatic numbers in the setting where larger subcubes of the hypercube are colored. We exhibit two new constructions which show that this problem is not a straightforward generalization of the edge coloring problem.Comment: 24 page

Polychromatic Colorings on the Integers

We show that for any set $S\subseteq \mathbb{Z}$, $|S|=4$ there exists a 3-coloring of $\mathbb{Z}$ in which every translate of $S$ receives all three colors. This implies that $S$ has a codensity of at most $1/3$, proving a conjecture of Newman [D. J. Newman, Complements of finite sets of integers, Michigan Math. J. 14 (1967) 481--486]. We also consider related questions in $\mathbb{Z}^d$, $d\geq 2$.Comment: 16 pages, improved presentatio

An Extremal Problem for Finite Lattices

For a fixed M x N integer lattice L(M,N), we consider the maximum size of a subset A of L(M,N) which contains no squares of prescribed side lengths k(1),...,k(t). We denote this size by ex(L(M,N), {k(1),...,k(t)}), and when t = 1, we abbreviate this parameter to ex(L(M,N), k), where k = k(1). Our first result gives an exact formula for ex(L(M,N), k) for all positive integers k, M, and N, where ex(L(M,N), k) = ((3/4) + o(1)) MN holds for fixed k and diverging M and N. Our second result identifies a subset A0 of L(M,N) of size at least (2/3)MN with the property that, for any integer k not divisible by three, A0 contains no squares of side length k. Our third result shows that |A0| is asymptotically best possible, in that for all positive integers M and N, we have ex(L(M,N), {1,2}) \u3c (2/3)MN + O(M+N). When M = 3m, our estimates on the error above render exact formulas for ex(L(3m,3), {1,2}) and ex(L(3m,6), {1,2})

Minimum Light Numbers in the σ\sigma -Game and Lit-Only σ\sigma -Game on Unicyclic and Grid Graphs

Consider a graph each of whose vertices is either in the ON state or in the OFF state and call the resulting ordered bipartition into ON vertices and OFF vertices a configuration of the graph. A regular move at a vertex changes the states of the neighbors of that vertex and hence sends the current configuration to another one. A valid move is a regular move at an ON vertex. For any graph $G,$ let $\mathcal{D}(G)$ be the minimum integer such that given any starting configuration $\bf x$ of $G$ there must exist a sequence of valid moves which takes $\bf x$ to a configuration with at most $\ell +\mathcal{D}(G)$ ON vertices provided there is a sequence of regular moves which brings $\bf x$ to a configuration in which there are $\ell$ ON vertices. The shadow graph $\mathcal{S}(G)$ of a graph $G$ is obtained from $G$ by deleting all loops. We prove that $\mathcal{D}(G)\leq 3$ if $\mathcal{S}(G)$ is unicyclic and give an example to show that the bound $3$ is tight. We also prove that $\mathcal{D}(G)\leq 2$ if $G$ is a two-dimensional grid graph and $\mathcal{D}(G)=0$ if $\mathcal{S}(G)$ is a two-dimensional grid graph but not a path and $G\neq \mathcal{S}(G)$

Does the lit-only restriction make any difference for the σ-game and σ+-game?

AbstractEach vertex in a simple graph is in one of two states: “on” or “off”. The set of all on vertices is called a configuration. In the σ-game, “pushing” a vertex v changes the state of all vertices in the open neighborhood of v, while in the σ+-game pushing v changes the state of all vertices in its closed neighborhood. The reachability question for these two games is to decide whether a given configuration can be reached from a given starting configuration by a sequence of pushes. We consider the lit-only restriction on these two games where a vertex can be pushed only if it is in the on state. We show that the lit-only restriction can make a big difference for reachability in the σ-game, but has essentially no effect in the σ+-game

Incidence of human brucellosis in the Kilimanjaro Region of Tanzania in the periods 2007-2008 and 2012-2014

Background: Brucellosis causes substantial morbidity among humans and their livestock. There are few robust estimates of the incidence of brucellosis in sub-Saharan Africa. Using cases identified through sentinel hospital surveillance and health care utilization data, we estimated the incidence of brucellosis in Moshi Urban and Moshi Rural Districts, Kilimanjaro Region, Tanzania, for the periods 2007–2008 and 2012–2014. Methods: Cases were identified among febrile patients at two sentinel hospitals and were defined as having either a 4-fold increase in Brucella microscopic agglutination test titres between acute and convalescent serum or a blood culture positive for Brucella spp. Findings from a health care utilization survey were used to estimate multipliers to account for cases not seen at sentinel hospitals. Results: Of 585 patients enrolled in the period 2007–2008, 13 (2.2%) had brucellosis. Among 1095 patients enrolled in the period 2012–2014, 32 (2.9%) had brucellosis. We estimated an incidence (range based on sensitivity analysis) of brucellosis of 35 (range 32–93) cases per 100 000 persons annually in the period 2007–2008 and 33 (range 30–89) cases per 100 000 persons annually in the period 2012–2014. Conclusions: We found a moderate incidence of brucellosis in northern Tanzania, suggesting that the disease is endemic and an important human health problem in this area