Questions tagged [extremal-set-theory]

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Families of subsets with pairwise symmetric differences of cardinality at most $k$

Let $X$ be an $n$-element set and $\mathcal{F} \subseteq P(X)$ such that for all $A, B \in \mathcal{F}$, $|A△B| \leq k$ where $A△B$ denotes the symmetric difference of $A$ and $B$. Suppose $|\mathcal{...
Francis Raj S's user avatar
8 votes
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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 ...
WangYao's user avatar
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5 votes
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161 views

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) $\forall S\in \mathcal{F}$, $|S|=a$....
Zihan Tan's user avatar
  • 151
4 votes
0 answers
114 views

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 ...
Seva's user avatar
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3 votes
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110 views

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 ...
Fabius Wiesner's user avatar
3 votes
0 answers
92 views

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: ...
Arun's user avatar
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3 votes
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132 views

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 ...
Seva's user avatar
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3 votes
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102 views

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 ...
JeremyKun's user avatar
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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 $...
SetFamilyStudent's user avatar
2 votes
1 answer
224 views

Union closed family of sets with at most a certain number of couples of sets with non-empty intersection

Is it possible to find a union closed family $\mathcal{F}$, $\emptyset \notin \mathcal{F}$, with $|\mathcal{F}| = n$ sets, such that there are at most: $$\left(1-\frac{1}{\left\lfloor \frac{n-1}{2} \...
Fabius Wiesner's user avatar
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 ...
Chris H's user avatar
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2 votes
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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 ...
GMB's user avatar
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1 vote
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44 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 $k$ ...
SetFamilyStudent's user avatar
1 vote
0 answers
120 views

On a generalisation of the EKR theorem

Let $n > k >t$ be positive integers, and let us assume $2k \leqslant n$. We denote the set of $k$-subsets of $[n]$ by $\mathcal{F}$. Let $C_1\subseteq \mathcal{F}$ be such that any two elements ...
Groups's user avatar
  • 369
1 vote
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Partitioning of a set family that avoids small intersections

Let $\mathcal{F}$ be the family of all $k$-element subsets of $[n]$. What is the smallest $\ell$ such that we can partition $\mathcal{F}$ into $\ell$ families $F_1,\dots,F_\ell$ with the property that ...
wandering_lambda's user avatar
1 vote
0 answers
103 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 ...
wandering_academic's user avatar
1 vote
0 answers
195 views

Additional condition to the Bollobas theorem (Sperner's therorem) in extremal set theory

The Bollobas'1965 theorem is the following: If $A_1,...,A_n$ and $B_1,...,B_n$ are two sequences of subsets of $X=\{1,...,r\}$ such that $A_i\cap B_j = \emptyset$ if and only if $i=j$, then $$\sum_{i=...
Paul Bub.'s user avatar
  • 105
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Integral representation of completely alternating homogeneous functionals on semi-lattice of continuous functions

For a long time I've been interested in G. Choquet seminal work "Theory of capacities" (Annales de l’institut Fourier, tome 5 (1954), p. 131-295). More precisely part 53 about integral ...
Vladimir B.'s user avatar