Intersecting families of convex cover order two

1975 ◽  
pp. 218-226
Author(s):  
Marilyn Breen
Keyword(s):  
Intersecting Families ◽  
Convex Cover

Weighted multiply intersecting families

2003 ◽  
Vol 40 (3) ◽  
pp. 287-291 ◽  
Author(s):  
Peter Frankl ◽  
Norihide Tokushige
Keyword(s):  
Intersecting Families

scholarly journals Uniform intersecting families with covering number four

1995 ◽  
Vol 71 (1) ◽  
pp. 127-145 ◽  
Author(s):  
Peter Frankl ◽  
Katsuhiro Ota ◽  
Norihide Tokushige
Keyword(s):  
Covering Number ◽  
Intersecting Families

scholarly journals Counting Intersecting and Pairs of Cross-Intersecting Families

2017 ◽  
Vol 27 (1) ◽  
pp. 60-68 ◽  
Author(s):  
PETER FRANKL ◽  
ANDREY KUPAVSKII
Keyword(s):  
Classical Result ◽  
Maximum Size ◽  
Extremal Combinatorics ◽  
Intersecting Family ◽  
Intersecting Families

A family of subsets of {1,. . .,n} is called intersecting if any two of its sets intersect. A classical result in extremal combinatorics due to Erdős, Ko and Rado determines the maximum size of an intersecting family of k-subsets of {1,. . .,n}. In this paper we study the following problem: How many intersecting families of k-subsets of {1,. . .,n} are there? Improving a result of Balogh, Das, Delcourt, Liu and Sharifzadeh, we determine this quantity asymptotically for n ≥ 2k+2+2$\sqrt{k\log k}$ and k → ∞. Moreover, under the same assumptions we also determine asymptotically the number of non-trivial intersecting families, that is, intersecting families for which the intersection of all sets is empty. We obtain analogous results for pairs of cross-intersecting families.

scholarly journals Multiply intersecting families of sets

2004 ◽  
Vol 106 (2) ◽  
pp. 315-326 ◽  
Author(s):  
Zoltán Füredi ◽  
Zsolt Katona
Keyword(s):  
Intersecting Families

scholarly journals Fractional L-intersecting Families

10.37236/7846 ◽  
2019 ◽  
Vol 26 (2) ◽  
Author(s):  
Niranjan Balachandran ◽  
Rogers Mathew ◽  
Tapas Kumar Mishra
Keyword(s):  
Intersecting Family ◽  
Intersecting Families

Let $L = \{\frac{a_1}{b_1}, \ldots , \frac{a_s}{b_s}\}$, where for every $i \in [s]$, $\frac{a_i}{b_i} \in [0,1)$ is an irreducible fraction. Let $\mathcal{F} = \{A_1, \ldots , A_m\}$ be a family of subsets of $[n]$. We say $\mathcal{F}$ is a fractional $L$-intersecting family if for every distinct $i,j \in [m]$, there exists an $\frac{a}{b} \in L$ such that $|A_i \cap A_j| \in \{ \frac{a}{b}|A_i|, \frac{a}{b} |A_j|\}$. In this paper, we introduce and study the notion of fractional $L$-intersecting families.

scholarly journals Most Probably Intersecting Hypergraphs

10.37236/4784 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
Shagnik Das ◽  
Benny Sudakov
Keyword(s):  
Initial Segment ◽  
Lexicographic Order ◽  
Structural Characterisation ◽  
Intersecting Families ◽  
Intersecting Hypergraphs

The celebrated Erdős-Ko-Rado theorem shows that for $n \ge 2k$ the largest intersecting $k$-uniform set family on $[n]$ has size $\binom{n-1}{k-1}$. It is natural to ask how far from intersecting larger set families must be. Katona, Katona and Katona introduced the notion of most probably intersecting families, which maximise the probability of random subfamilies being intersecting.We consider the most probably intersecting problem for $k$-uniform set families. We provide a rough structural characterisation of the most probably intersecting families and, for families of particular sizes, show that the initial segment of the lexicographic order is optimal.

scholarly journals Cross L-intersecting families on set systems

2010 ◽  
Vol 310 (4) ◽  
pp. 720-726 ◽  
Author(s):  
Jiuqiang Liu ◽  
Xiaodong Liu
Keyword(s):  
Set Systems ◽  
Intersecting Families
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