本文介绍一下集系极值问题中一个非常著名的定理: Erdos-Ko-Rado 定理. 这个定理给出了 相交系 的紧上界. 设 \mathcal {F} 是 \ {1,2,...,n\} 的一些子集构成的集族, 我们称 \mathcal {F} 是一个相交系：如果对任意两个集合 A,B\in \mathcal {F} , 有 A\cap B\ne \emptyset . 我们考虑 [n] 的 一些 k -元子集构成的相交系最大能有多大. Erdos, 柯召，Rado证明了下面的定理: Theorem 1 (Erdos-Ko-Rado): 设 n\geq 2k ， 那么 [n] 的k元子集构成的相交系的规模至多只能是 \binom {n-1} {k-1} .

定理1.1 (EKR 定理 [4]) 假设n 和k 为正整数且n > 2k. 对于任意交族F ⊂ (n] k), 有以下结论: |F| 6 (n−1 k −1). (1.1) 此外, 如果n > 2k, 则(1.1) 中的等式成立当且仅当F 由包含[n] 的某一个元素的所有k 元子集构成. 例如, A1,1 是一个交族并且达到了(1.1) 中的上界.

2020年7月2日 · A novel application of the theory of buildings and Iwahori-Hecke algebras is developed to prove sharp upper bounds for EKR-sets of flags. In this framework, we can reprove and generalize previous upper bounds for EKR-problems in projective and polar spaces.

The EKR theorem is follows by carefully choosing the intersection properties and adding extra polynomials. We also prove generalizations for non-uniform families with various intersection conditions.

Erdős–Ko–Rado (EKR) type theorems yield upper bounds on the sizes of families of sets, subject to various intersection requirements on the sets in the family. Stability versions of such theorems assert that if the size of a family is close to the maximum possible size, then the family itself must be close (in some appropriate sense) to a ...

The intersection density rho (G) is defined to be the ratio |S|/|G_w|, where S is a maximum intersecting subset, and G is said to have EKR-property if rho (G)=1. We define a fixer of G to be an...

In this paper we study k-uniform intersecting families from the point of view of the minimum vertex degree. There have been work on intersecting families with maximum degree conditions. Department of Math and CS, Emory University, Atlanta, GA 30322. Email: [email protected]. Research supported in part by Simons Collaboration Grant.

The authors introduce tools commonly used in algebraic graph theory and show how these can be used to prove versions of the EKR Theorem. Topics include association schemes, strongly regular graphs, the Johnson scheme, the Hamming scheme and the Grassmann scheme.

组合学研究的一类重要问题是极值组合问题，EKR定理（Erdős-Ko-Rado 定理，其中Ko是中国数学家柯召）是极值组合研究中的一个核心定理，产生了深远的影响。

2005年12月6日 · The classical Erd\H os-Ko-Rado theorem states that if $k\le\floor {n/2}$ then the largest family of pairwise intersecting -subsets of is of size . A family of subsets satisfying this pairwise intersecting property is called an EKR family.