All Questions
Tagged with co.combinatorics extremal-set-theory
40 questions
17
votes
5
answers
1k
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 ...
- 114k
15
votes
1
answer
717
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 ...
- 23k
14
votes
1
answer
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 $c_2$ ...
13
votes
2
answers
375
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 ...
- 32.1k
12
votes
2
answers
425
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 ...
- 3,004
11
votes
3
answers
636
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
...
- 237
9
votes
0
answers
335
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{...
8
votes
1
answer
1k
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 ...
- 109k
8
votes
0
answers
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 ...
- 393
7
votes
1
answer
774
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 ...
- 171
6
votes
2
answers
392
views
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 ...
- 135
6
votes
2
answers
489
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 \...
- 4,229
6
votes
1
answer
277
views
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 ...
- 419
6
votes
3
answers
511
views
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 ...
- 63
6
votes
1
answer
613
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 ...
- 30.3k
5
votes
1
answer
254
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 ...
- 30.1k
5
votes
1
answer
221
views
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 ...
- 19.1k
4
votes
1
answer
220
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 $MH(...
- 3,004
4
votes
1
answer
137
views
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 ...
- 43
4
votes
1
answer
864
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 ...
- 23k
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 ...
- 23k
3
votes
1
answer
215
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 ...
- 419
3
votes
1
answer
288
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 $k$ ...
- 131
3
votes
1
answer
275
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 ...
3
votes
0
answers
124
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 ...
- 1,000
3
votes
0
answers
95
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: ...
- 31
3
votes
0
answers
133
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 ...
- 23k
3
votes
0
answers
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 ...
- 726
2
votes
1
answer
197
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. ...
- 1,215
2
votes
1
answer
131
views
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}\...
- 19.1k
2
votes
1
answer
144
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 ...
- 43
2
votes
1
answer
877
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 $...
2
votes
1
answer
142
views
Bounds for ground set of Steiner system (inverse EKR style problem)
Imagine we have $r$ subsets of a ground set $S$, each of size $k$, such that each set of size $l$ is contained in at most one of the $r$ sets. What can we say about the minimum value of $|S|$? I am ...
- 83
2
votes
0
answers
99
views
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 $...
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
137
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 ...
- 379
1
vote
0
answers
79
views
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 ...
- 135
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 ...
1
vote
0
answers
204
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=...
- 105
0
votes
1
answer
181
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 $E \subseteq X$ ...
- 6,962