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Erd˝ os asked in 1962 about the value of f (n, k, l), the minimum number of k-cliques in a graph with order n and independence number less than l. The case (k, l) = (3, 3) was solved by Lorden. Here we solve the problem (for all large n) for (3, l) with 4 ≤ l ≤ 7 and (k, 3) with 4 ≤ k ≤ 7.

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)

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 F 3,2 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 [9] 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)

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