Emil R. Vaughan

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Given an r-graph H on h vertices, and a family F of forbidden subgraphs, we define exH(n,F) to be the maximum number of induced copies of H in an F-free r-graph on n vertices. Then the Turán H-density of F is the limit πH(F) = lim n→∞ exH(n,F)/ ( n h ) . This generalises the notions of Turán density (when H is an r-edge), and inducibility (when F is empty).(More)
The k-domination number of a graph is the cardinality of a smallest set of vertices such that every vertex not in the set is adjacent to at least k vertices of the set. We prove two bounds on the k-domination number of a graph, inspired by two conjectures of the computer program Graffiti.pc. In particular, we show that for any graph with minimum degree at(More)
Gerechte designs are a specialisation of latin squares. A gerechte design is an n×n array containing the symbols {1, . . . , n}, together with a partition of the cells of the array into n regions of n cells each. The entries in the cells are required to be such that each row, column and region contains each symbol exactly once. We show that the problem of(More)
Let us give some definitions first. As usual, a graph G is a pair (V (G), E(G)), where V (G) is the vertex set and the edge set E(G) consists of unordered pairs of vertices. An isomorphism between graphs G and H is a bijection f : V (G) → V (H) that preserves edges and non-edges. For a graph G, let G = ( V (G), ( V (G) 2 ) \E(G) ) denote its complement and(More)
Guessing games for directed graphs were introduced by Riis [8] for studying multiple unicast network coding problems. It can be shown that protocols for a multiple unicast network can be directly converted into a strategy for a guessing game. The performance of the optimal strategy for a graph is measured by the guessing number, and this number can be(More)
Ocular pain is often treated with systemic analgesics, which are associated with some undesirable side effects. Because nonsteroidal anti-inflammatory drugs are thought to be peripheral analgesics, we decided to evaluate the local analgesic effect of flurbiprofen. After an initial study in 29 patients demonstrated that 0.03% flurbiprofen ophthalmic solution(More)
Given a family of 3-graphs F , we define its codegree threshold coex(n,F) to be the largest number d = d(n) such that there exists an n-vertex 3-graph in which every pair of vertices is contained in at least d 3-edges but which contains no member of F as a subgraph. Let F3,2 be the 3-graph on {a, b, c, d, e} with 3-edges abc, abd, abe and cde. In this(More)
Guessing games for directed graphs were introduced by Riis for studying multiple unicast network coding problems. In a guessing game, the players toss generalised dice and can see some of the other outcomes depending on the structure of an underlying digraph. They later guess simultaneously the outcome of their own die. Their objective is to find a(More)