A family F⊂[n] k is called an intersecting family if F∩ F′≠ 0̸ for all F, F′∈ F. If∩ F≠ 0̸
then F is called a star. The diversity of an intersecting family F is defined as the minimum …

For a family of subsets of an n-element set, its matching number is the maximum number of
pairwise disjoint sets. Families with matching number 1 are called intersecting. The famous …

Let S be an n-element set and F⊂(S k) an intersecting family. Improving earlier results it is
proved that for n> 72 k there is an element of S that is contained in all but (n− 3 k− 2) …

For an n-element set X let X k be the collection of all its k-subsets. Two families of sets A and
B are called cross-intersecting if A∩ B≠ 0̸ holds for all A∈ A, B∈ B. Let f (n, k) denote the …

Refuting conjectures in extremal combinatorics via linear programming - ScienceDirect Skip to
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For a family $\mathcal {F}\subset\binom {[n]}{k} $ and two elements $ x, y\in [n] $ define
$\mathcal {F}(\bar {x},\bar {y})=\{F\in\mathcal {F}\colon x\notin F,\y\notin F\} $. The double …

Let F⊆[n] r be an intersecting family of sets and let Δ (F) be the maximum degree in F, ie, the
maximum number of edges of F containing a fixed vertex. The diversity of F is defined as d …

Given a family F⊂ 2 [n], its diversity is the number of sets not containing an element with the
highest degree. The concept of diversity has proven to be very useful in the context of k …

Two families $\mathcal {F} $ and $\mathcal {G} $ are called cross-intersecting if for every $
F\in\mathcal {F} $ and $ G\in\mathcal {G} $, the intersection $ F\cap G $ is non-empty. It is …

We consider $ k $-graphs on $ n $ vertices, that is, $\mathcal {F}\subset\binom {[n]}{k} $. The
recent advances of Keller and Lifshitz gave a new impetus to extend the validity of …