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Questions tagged [extremal-set-theory]

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6
votes
2answers
233 views

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
votes
0answers
36 views

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
vote
1answer
77 views

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
votes
1answer
99 views

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 ...
3
votes
1answer
203 views

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 $k$ ...
12
votes
2answers
257 views

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 ...
3
votes
0answers
95 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) $\forall S\in \mathcal{F}$, $|S|=a$....
4
votes
1answer
79 views

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 $...
8
votes
1answer
371 views

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 ...
2
votes
1answer
144 views

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. ...
1
vote
0answers
149 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=...
3
votes
0answers
107 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 ...
4
votes
0answers
94 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 ...
6
votes
1answer
304 views

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 ...
9
votes
0answers
200 views

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{...
2
votes
1answer
196 views

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 $...
3
votes
0answers
92 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 ...
3
votes
1answer
159 views

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 ...
12
votes
2answers
356 views

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 ...
0
votes
1answer
141 views

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 $E \subseteq X$ ...
5
votes
1answer
194 views

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 ...
7
votes
0answers
1k views

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 ...
15
votes
5answers
955 views

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
votes
2answers
856 views

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 ...
4
votes
1answer
558 views

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 ...
3
votes
1answer
1k views

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
votes
1answer
588 views

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
votes
1answer
243 views

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 ...
14
votes
1answer
1k views

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
votes
2answers
391 views

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 \...
7
votes
1answer
503 views

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 ...