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Some Erdős-Ko-Rado type extremal properties of families of vectors from {-1; 0; 1}n are considered.

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Abstract  

We determine a class of triple systems such that each must occur in a triple system with uncountable chromatic number that omits
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(the unique system consisting of two triples on four vertices). This class contains all odd circuits of length ≧ 7. We also show that consistently there are two finite triple systems such that they can separately be omitted by uncountably chromatic triple systems but not both.
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An upper bound is given on the size of a k-fan-free 3-graph, and an infinite family reaching this bound is also described.

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Let forb(m, F) denote the maximum number of columns possible in a (0, 1)-matrix A that has no repeated columns and has no submatrix which is a row and column permutation of F. We consider cases where the configuration F has a number of columns that grows with m. For a k × l matrix G, define s · G to be the concatenation of s copies of G. In a number of cases we determine forb(m, m α · G) is Θ(m k). Results of Keevash on the existence of designs provide constructions that can be used to give asymptotic lower bounds. An induction idea of Anstee and Lu is useful in obtaining upper bounds.

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Abstract

We develop a model of scientific creativity and test it in the field of rare diseases. Our model is based on the results of an in-depth case study of the Rett Syndrome. Archival analysis, bibliometric techniques and expert surveys are combined with network analysis to identify the most creative scientists. First, we compare alternative measures of generative and combinatorial creativity. Then, we generalize our results in a stochastic model of socio-semantic network evolution. The model predictions are tested with an extended set of rare diseases. We find that new scientific collaborations among experts in a field enhance combinatorial creativity. Instead, high entry rates of novices are negatively related to generative creativity. By expanding the set of useful concepts, creative scientists gain in centrality. At the same time, by increasing their centrality in the scientific community, scientists can replicate and generalize their results, thus contributing to a scientific paradigm.

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