Abstract. 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 paper, we give two proofs that coex(n, {F 3,2 }) = + o(1) n, the first by a direct combinatorial argument and the second via a flag algebra computation. Information extracted from the latter proof is then used to obtain a stability result, from which in turn we derive the exact codegree threshold for all sufficiently large n: coex(n, {F 3,2 }) = n/3 − 1 if n is congruent to 1 modulo 3, and n/3 otherwise. In addition we determine the set of codegree-extremal configurations for all sufficiently large n.
In "The structure of hereditary properties and colourings of random graphs" [Combinatorica 20(2) (2000), 173-202], Bollobás and the second author studied the probability of a hereditary property P in the probability space G(n, p). They found simple properties that closely approximate P in this space, and using these simple properties they determined the P-chromatic number of random graphs.In this note we point out that the analysis of hereditary properties in G(n, p) can be made more exact by means of the extremal properties of 2-coloured multigraphs, and we illustrate some cases in which the probability of P can thereby be calculated. At the same time we correct a mistake in the cited paper.
A 3-graph is said to contain a generalized 4-cycle if it contains 4 edges A, B, C, D such that A ∩ B=C ∩ D =∅ and A ∪ B=C ∪ D. We show that a 3-graph in which every pair of vertices is contained in at least 4 edges must contain a generalized 4-cycle. When the number of vertices, n, is equivalent to 1 or 5 modulo 20, this result is optimum, in the sense that for such n there are 3-graphs where every pair of vertices is contained in 3 edges but which do not contain a generalized 4-cycle.
The role of Pope Pius XI in dealing with the spreading of Nazi racism is one that has been overshadowed by his controversial successor's actions during the Holocaust. This historical narrative fails to recognize the achievements of this relatively centrist pope in an era of extremes. In fact, Pope Pius XI played a crucial part in laying the foundation for Christian and Jewish resistance against Nazi racist ideologies and cruelty. Indeed, his actions are in part a makeup of his life's mission. Thus, by beliefs, Pope Pius XI can be seen both as an opponent to racial antisemitism and a man attempting to keep the Church safe while caught in a world of rapid change due to the ideal of modernity and the spreading of Communism and Fascism.
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