Abstract
A family $\mathcal F$ of $k$-element subsets of the $n$-element set $[n]$ is called \emph{intersecting} if $F \cap F'\neq \emptyset$ for all $F, F' \in \mathcal F$. In 1961 Erdős, Ko and Rado showed that $|\mathcal F| \leq {n - 1\choose k - 1}$ if $n \geq 2k$. Since then a large number of resultső providing best possible upper bounds on $|\mathcal F|$ under further restraints were proved. The paper of Li et al. is one of them. We consider the restricted universe $\mathcal W = \left\{F \in {[n]\choose k}: |F \cap [m]| \geq \ell \right\}$, $n \geq 2k$, $m \geq 2\ell$ and determine $\max |\mathcal F|$ for intersecting families $\mathcal F \subset \mathcal W$. Then we use this result to solve completely the problem considered by Li et al.
Highlights
Let n, k be positive integers, n 2k
A family F is called intersecting if F ∩ F = ∅ holds for all F, F ∈ F
The Erdos–Ko–Rado Theorem was the origin of a lot of research, there are many papers strengthening, generalizing or extending it
Summary
Let n, k be positive integers, n 2k. Let [n] = {1, 2, . . . , n} be the standard n-element set and [n] k the collection of all its k-element subsets. Subsets F of [n] k are called k-uniform families. A family F is called intersecting if F ∩ F = ∅ holds for all F, F ∈ F. The simplest example of a large intersecting family is the star:. The classical Erdos–Ko–Rado Theorem [EKR] states that no k-uniform intersecting family can surpass |S|. Motivated by a paper of Li, Chen, Huang and Li [LCHL] we consider the following problem. Let g(n, k, m, ) denote the maximum of |F| where F ⊂ W(n, k, m, ) is intersecting. Let us define the restricted star R = R(n, k, m, ) by R = S ∩ W, that is, R= R∈. : 1 ∈ H, m − k + 1 |H ∩ [m]| k − 1
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