Questions tagged [extremal-set-theory]
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47
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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 ...
5
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1
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194
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How many base elements can a sunflower-free system have?
A collection of $t$ sets $A_i$ is called a t-sunflower if $A_i \cap A_j = Z $ for all $i \neq j$ for some fixed $Z$. A well-known conjecture of Erdős and Rado says that there is a constant $C_t$ such ...
2
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224
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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} \...
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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 $...
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44
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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
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120
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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 ...
6
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3
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506
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Partitioning the 3-sets of [n]={1,...,n} into families
Let $F_1,...,F_m$ be a partition of the 3-element subsets of $[n]$ into families such that no three subsets in any one family $F_i$ are all contained in one 4-element subset of $[n]$. What is the ...
2
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1
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126
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Turán density of hypergraphs with very few edges
As usual, for an $r$-uniform hypergraph $G$, denote by $ex_r(n,G)$ the maximum number of edges an $r$-uniform, $G$-free hypergraph on $n$ vertices can have, and let $\lim \frac{ex_r(n,G)}{\binom nr}\...
6
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257
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A Sauer-Shelah-like lermma for prefix tree
I proved a variant of the Sauer-Shelah lemma and I was wondering if something like that is already known.
Let $S \subseteq \{0,1\}^n $. We say that a set of coordinates $K \subseteq [n]$ is shattered ...
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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 ...
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119
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On the number of disjoint subsets of a large set families
Let $[n] := \{1,\dots,n\}$, for some large integer $n$, and let $\mathcal{F}$ be a family of 2-element subsets of $[n]$.
The famous Erdös-Ko-Rado (EKR) theorem says that if $|\mathcal{F}| > {n - 1 ...
6
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2
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385
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Coloring of a graph representing the power set
For a positive integer $n$, let $\mathcal{P}$ be the power set of $[n]$. Consider the graph $G$ with $\mathcal{P}$ as its vertex set, and, for $S_1,S_2 \in \mathcal{P}$, the edge $(S_1,S_2)$ exists ...
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77
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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 ...
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103
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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 ...
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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: ...
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594
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Domination problem with sets
For nearly two years, I have been struggling with the next task I have already published on MSE, but unfortunately with no respond.
Let $M$ be a non-empty and finite set, $S_1,...,S_k$ subsets
...
2
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87
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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 ...
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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 ...
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1
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137
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Number of distinct 3-sets formed as subsets of a 4-uniform set family
Let $\mathcal{F} \subseteq \binom{[n]}{4}$ be a collection of size-4 subsets of $\{1,\ldots,n\}$, i.e., a 4-uniform set family.
Let $\mathcal{F'}$ be the collection of size-3 sets for which a ...
2
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1
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139
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Abundance in union closed families
For any finite set $S$ and every partition $S_1, \dots, S_n$ of $S$, let $P(S_1, \dots, S_n)$ be the family consisting of all possible unions of $S_1, \dots, S_n$. Clearly, $P(S_1, \dots, S_n)$ is a ...
2
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193
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Number of members of a separating union-closed family whose universe has given cardinality
If I'm not wrong, it is easy to prove the following statement:
If $n \leq 4$ is a natural number, if $\mathcal{F}$ is a union-closed family of non-empty sets, if the universe of $\mathcal{F}$ (i.e. ...
3
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1
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272
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the size of a down-set?
I'm reading a research article lately, and got confused about a question.
So, the fundamental theorem of Kruskal and Katona states that if each set in a given set system $\mathcal{A}$ has $k$ ...
13
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2
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354
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Set family $\mathcal{F}$ such that for all $A,B,C \in \mathcal{F}$ both $A \cap B \not \subseteq C$ and $C \not \subseteq A \cup B $
This question initially arose out of a question in asymptotic matroid theory. The matroid question has since been answered in a different way, but the extremal set theory question remains unanswered ...
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161
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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$....
4
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1
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214
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Maximal number of perfect matchings that pairwise form a Hamiltonian cycle
Definition: Let $MH(n)$ be the maximal number of perfect matchings (1-regular graphs) on $n$ vertices where the union of any two perfect matchings is a Hamiltonian cycle.
Question: Is it true that $MH(...
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On a result of Frankl and Wilson
In the paper 'Intersection theorems with geometric consequences' (Combinatorica 1981) P. Frankl and R. M. Wilson consider families $\mathcal{F}$ of $k$-subsets of $\{1,\dots,n\}$ with the restriction ...
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Cyclic subgroups of finite abelian groups
I learned from MO Subgroups of a finite abelian group that the problem of enumerating subgroups (not up to isomorphism) of finite abelian groups is a difficult one.
Are there simple formulas if one ...
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195
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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=...
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132
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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 ...
12
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2
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418
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Set system with different differences
What is the maximal number of sets in a set system $\mathcal{A}$ of subsets of an $n$ element set such that for every $i \neq j $ and $A_i,A_j \in \mathcal{A}$ the difference $A_i \setminus A_j$ is ...
4
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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 ...
6
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578
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Given k, what is the minimum n such that n choose n/2 is greater than k? [closed]
I'm not an expert in combinatorics, but it sometimes comes up in my research with students in computer science (which is already pretty far away from my speciality of abstract homotopy theory). I just ...
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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{...
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Combinatorics-the maximum number of subsets with a given property
Let $X$ be a set with $n$ elements. I would like to know the maximum number of subsets of $X$ such that the number of elements in the symmetric difference between any two of these subsets is at most $...
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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 ...
3
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1
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Minimal family of k-sets containing all t-sets
Let $n \ge k \ge t \in \mathbb{N}$, and consider a universe $U$ of size $n$. Let $\mathcal{F}$ be a family of $k$-subsets of $U$, such that every $t$-subset of $U$ is contained in at least one member ...
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180
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Generalized Helly theorem for $t$-intersecting families
Given a family $\mathcal{F}$ of sets over ground set $X$, let $\tau(\mathcal{F})$ be the transversal number (aka blocking number), that is the cardinality of the smallest set of points $E \subseteq X$ ...
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Can a partition free family in $2^{[n]}$ always be enlarged to one of size $2^{n-1}$?
Let $\left[ n \right]=\{{1,2,\cdots,n\}}$ and call a family $\mathcal{F} \subset 2^{\left[n\right]}$ partition-free if it does not contain any partition of $\left[n\right]$. A recent question asked ...
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Upper bound for the size of a $k$-uniform $s$-wise $t$-intersecting set system
Given integers $n \geq k \geq t \geq 1$ and an integer $s$, let $m(n,k,s,t)$ denote the maximum size of a family $\mathcal F$ of $k$-subsets of $[n]$, i.e. $\mathcal F \subseteq \binom{[n]}{k}$, such ...
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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 ...
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Optimal bounds for an alternating sum on a downset
Let $n$ be a natural number, and consider the discrete cube $2^{[n]} := \{ A: A \subset \{1,\ldots,n\}\}$ consisting of all subsets of the $n$-element set $[n] := \{1,\ldots,n\}$. Define a downset in ...
4
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830
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Intersecting Hamming spheres: is $|A\stackrel k+E|\ge|A|$?
Since my original posting some ten days ago, I discovered an amazing
example which changed significantly my perception of the problem.
Accordingly, the whole post got re-written now.
The most general ...
4
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1
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Minimal generation for finite abelian groups
Let $G$ be a finite abelian group. I know of two ways of writing it as a direct sum of cyclic groups:
1) With orders $d_1, d_2, \ldots, d_k$ in such a way that $d_i|d_{i+1}$,
2) With orders that are ...
15
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1
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705
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The hypercube: $|A {\stackrel2+} E| \ge |A|$?
I have a good motivation to ask the question below, but since the post is
already a little long, and the problem looks rather natural and appealing
(well, to me, at least), I'd rather go straight to ...
3
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1
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268
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Lower bounding the maximum size of sets in a set family with union promise
The following problem has come up while working on the relationship between certificate and randomized decision tree complexities of boolean functions. However, I think it is of interest by itself and ...
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How to keep subsets disjoint?
Given positive integers $n$ and $k\le 2^n$, how to choose a subset $C\subset\{0,1\}^n$ of size $|C|=k$ to maximize the number of pairs $(c_1,c_2)\in C\times C$ with the supports of $c_1$ and $c_2$ ...
6
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459
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What is the largest family F of subsets of [n] for which any two distinct sets A and B in F have an intersection of size at most min(|A|,|B|)/2?
This problem arose in the study of Latin squares with a large number of subsquares, although it appears interesting in its own right.
Question: What is the maximum cardinality of a family $F \...