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Showing 1 to 5 of 5 for “"Intersecting families"”.
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Cliques in block graphs of designs and orthogonal arrays
The Erdos-Ko-Rado [EKR] Theorem for intersecting families is a fundamental result in combinatorics, particularly in extremal set theory. This theorem not only establishes an upper bound on the size of the largest intersecting family but also characterizes the families that attain this bound—–these …
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Viewing extremal and structural problems through a probabilistic lens
… 2 is determining the typical structure of $t$-intersecting families in various settings and enumerating such systems. The analogous sparse random versions of our extremal results are also obtained. The proofs follow the same general framework, in each case using a version of the Bollobás …
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Combinatorics of finite sets
… every subset of a member of I is also in I. An intersecting family S in 2$\sp{\lbrack n\rbrack }$ is called a star if there exists an element of (n) belonging to every member of S, and it is a 1-star if the intersection of every two members of I is exactly that element. Chvatal conjectured that …
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Some Results in Combinatorics and Combinatorial Geometry
… geometry. In Chapter 2, we discuss union-closed families. For a given number of k-sets, how should we choose them so as to minimise the union-closed family that they generate? In this chapter we show that, if $\mathcal{A}$ is a family of k-sets of size $\binom{t}{k}$, and t is sufficiently large, …