All Questions
Tagged with extremal-set-theory extremal-combinatorics
12 questions with no upvoted or accepted answers
8
votes
0
answers
1k
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The state of art of the sunflower lemma
I am interesting in the sunflower system and its applications in computer science.
Given a Universe $U$ and a collection of $k$ sets $A_i$ is called a k-sunflower system if $A_i \cap A_j = Y $ for ...
- 393
5
votes
0
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163
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A Combinatorial Problem on Extremal Set Theory
Given a ground set $[n]$, under what condition of parameters $a,b,c$ does a family of subsets $\mathcal{F}\subseteq 2^{[n]}$ with the following property exist?
(i) $\forall S\in \mathcal{F}$, $|S|=a$....
- 151
4
votes
0
answers
114
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Kruskal-Katona for homocyclic groups?
I need a version of the Kruskal-Katona theorem (better still, of the Lovasz "approximate" version thereof) for the elementary abelian / homocyclic groups, in the following spirit:
What is the ...
- 23k
3
votes
0
answers
124
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Minimum number of couples of sets with non-empty intersection in a union closed family
Every union closed family $\mathcal{F}$, $\emptyset \notin \mathcal{F}$, with $|\mathcal{F}| = n$ sets, must have at least $\frac{2}{3}\binom{n}{2}$ unordered couples of sets with at least one element ...
- 1,000
3
votes
0
answers
95
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Minimum number of partitions of a set such that the same pair must occur in a block in at least half of them
For positive integers $k$ and $n$, let ${S} = \{1,\dots,k\ n\}$. Consider $\ell \ge 3$ partitions $P_1,\dots,P_\ell$ of ${S}$, where each $P_i$ splits ${S}$ into $n$ blocks all of size $k$.
Question: ...
- 31
3
votes
0
answers
133
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Kruskal-Katona for multisets?
Following Fedor Petrov's remarks, here is a "set-theoretic version" of the
question I asked a while ago.
For integer $n\ge 1$, denote by $\mathcal M_n$ the family of all (finite)
multisets with the ...
- 23k
3
votes
0
answers
102
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What is the maximal number of partitions with this maximal intersection property?
Let $X = \{ 1, \dots, n = sk \}$ be a finite set. Let $\mathscr P, \mathscr Q$ be equi-partitions of $X$ into $k$ sets of size $s$. Denote by $V(\mathscr P, \mathscr Q)$ the maximum size of ...
- 726
2
votes
0
answers
99
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A variant of the social golfer problem and the kirkman schoolgirl problem
I came across the following simple question that seems to be open:
Let $U$ be a set of $n$ elements.
Let $P_1$ be a partition of $U$ into $k\le n$ "blocks" (i.e. disjoint subsets) and let $...
2
votes
0
answers
87
views
Restricted Erdos-Ko-Rado theorem reference
Consider a family $\mathfrak{F}$ of $k$ element subsets of $\{1,2,..,n\}$, where $n\geq 2k$, such that any two members of $\mathfrak{F}$ have nonempty intersection. The Erdos-Ko-Rado theorem asserts ...
- 1,949
2
votes
0
answers
59
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Is the finite projective plane stable as an extremal set system?
Let $\Sigma$ be a set of $|\Sigma| = n$ subsets of the universe $[n]$, each of size $k$, with the property that any two of these subsets intersect on at most one element. It is easy to see that the ...
- 1,389
1
vote
0
answers
45
views
How small must partitions be to ensure overlapping blocks?
Consider the set family $F$ of all $t$-element subsets of $[n]$, for some positive integer $n$.
Let $P_1$ be a partition of $F$ into $k$ blocks.
Let $P_2 \ne P_1$ be another partition of $F$ into $k$ ...
1
vote
0
answers
104
views
Number of intersections that must occur in any partition of a given size
Let $\mathcal{S}$ be the set of all $n$-element subsets of $[2n]:=\{1,\dots,2n\}$.
Consider a partition $\mathcal{P}$ of $\mathcal{S}$ into $m$ blocks $P_1,\dots,P_m$, where all except at most one of ...
