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A family of sets is said to be emph{symmetric} if its automorphism group is transitive, and emph{intersecting} if any two sets in the family have nonempty intersection. Our purpose here is to study the following question: for $n, kin mathbb{N}$ with $k le n/2$, how large can a symmetric intersecting family of $k$-element subsets of ${1,2,ldots,n}$ be? As a first step towards a complete answer, we prove that such a family has size at most [expleft(-frac{c(n-2k)log n}{k( log n - log k)} right) binom{n}{k},] where $c > 0$ is a universal constant. We also describe various combinatorial and algebraic approaches to constructing such families.
We shall be interested in the following Erdos-Ko-Rado-type question. Fix some subset B of [n]. How large a family A of subsets of [n] can we find such that the intersection of any two sets in A contains a cyclic translate (modulo n) of B? Chung, Grah
A family of perfect matchings of $K_{2n}$ is $t$-$intersecting$ if any two members share $t$ or more edges. We prove for any $t in mathbb{N}$ that every $t$-intersecting family of perfect matchings has size no greater than $(2(n-t) - 1)!!$ for suffic
Let $mathcal{F}$ and $mathcal{G}$ be two $t$-uniform families of subsets over $[k] = {1,2,...,k}$, where $|mathcal{F}| = |mathcal{G}|$, and let $C$ be the adjacency matrix of the bipartite graph whose vertices are the subsets in $mathcal{F}$ and $mat
The extremal problems regarding the maximum possible size of intersecting families of various combinatorial objects have been extensively studied. In this paper, we investigate supersaturation extensions, which in this context ask for the minimum num
A family of perfect matchings of $K_{2n}$ is $intersecting$ if any two of its members have an edge in common. It is known that if $mathcal{F}$ is family of intersecting perfect matchings of $K_{2n}$, then $|mathcal{F}| leq (2n-3)!!$ and if equality h