All Questions
Tagged with co.combinatorics extremal-combinatorics
29 questions
8
votes
1
answer
1k
views
Looking for another difficult case for the union-closed sets conjecture
[Now crossposted at math.stackexchange.]
Consider a union-closed family $\mathcal{F}$ of $n$ finite sets with $\mathcal{F} \not = \{ \emptyset \}$.
Let $\mathcal{H} \subseteq \mathcal{F}$ be the ...
- 990
4
votes
2
answers
557
views
Conjecture about union-closed families of sets - attempt 3
Version 2 of the conjecture was disproved. In this version 3 of the conjecture I am adding a further requirement to obtain from $\mathcal{H}$ a "minimal" family $\mathcal{G}$.
I have already ...
- 990
10
votes
1
answer
401
views
How many positions of a tiling polygon can occur simultaneousy?
Let $T$ be a polygon which tiles the plane. For an instance of $T$ (mirrored or not), call the set of its translates a position of $T$.
My question:
How many different positions can occur in ...
- 13.4k
4
votes
1
answer
442
views
Conjecture about union-closed families of sets - attempt 2
[Update: this one has been disproved, but I have started conjecture attempt 3]
My previous question had an error, I am sorry for that. The limit $\lceil n/2 \rceil$ must be replaced with $\lceil (n+1)/...
- 990
3
votes
0
answers
184
views
Matrices with only two different entries and maximal determinant
Define $\mathcal M_n$ as the set of all $n\times n$ matrices of full rank with each entry either 1 or $x$.
I am interested in how big the determinant of such a matrix can be. For this, we define in a ...
- 13.4k
0
votes
0
answers
115
views
Simpler lower bound for couples of disjoint sets
This is similar to a previous question, but simpler, I suppose.
Let $\mathcal{B}$ be the family of all subsets of $[n]=\{1,2,\ldots,n\} $ of size $2$. Let $\mathcal{F} = \{\mathcal{A}_1,\ldots,\...
- 990
56
votes
1
answer
3k
views
Intersecting family of triangulations
Let $\cal T_n$ be the family of all triangulations on an $n$-gon using $(n-3)$ non-intersecting diagonals. The number of triangulations in $\cal T_n$ is $C_{n-2}$ the $(n-2)$th Catalan number. Let $\...
- 24.7k
26
votes
2
answers
2k
views
What is the best lower bound for 3-sunflowers?
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 Erdos and Rado says that there is a constant $C_t$ such ...
- 18.7k
26
votes
3
answers
907
views
What is the smallest size of a shape in which all fixed $n$-polyominos can fit?
Let $n$ be an integer and consider all fixed $n$-polyominos, i.e., without rotation or reflection. I am interested in finding a shape in which all polyominos can embed. (It is OK if multiple ...
- 431
23
votes
3
answers
3k
views
Cauchy-Schwarz proof of Sidorenko for 3-edge path (Blakley-Roy inequality)
Is there a "Cauchy-Schwarz proof" of the following inequality?
Theorem. Given $f \colon [0,1]^2 \to [0,1]$, one has
$$
\int_{[0,1]^4} f(x,y)f(z,y)f(z,w) \, dxdydzdw \geq \left(\int_{[0,1]^2} f(x,y) \,...
- 986
14
votes
1
answer
633
views
Minimal "sumset basis" in the discrete linear space $\mathbb F_2^n$
For a set $C\subseteq \mathbb F_2^n$, let $2C=C+C:=\{\alpha+\beta\colon \alpha,\beta\in C\}$.
I want to find $C$ of the smallest possible size such that $2C=\mathbb F_2^n$. Let $m(n)$ be the size of a ...
- 197
12
votes
2
answers
742
views
Looking for a counterexample for a conjecture about union-closed families of sets
(I have posted a corrected version of this question. The limit $\lceil n/2 \rceil$ must be replaced with $\lceil (n+1)/2 \rceil$.)
I have already asked basically the same question here, but now I have ...
- 990
11
votes
2
answers
788
views
Blocking sets in three dimensional finite affine spaces
What is the smallest possible size of a set of points in $\mathbb{F}_q^3$ which intersects (blocks) every line?
Clearly the union of three affine hyperplanes that intersect in a singleton, say $x = 0,...
- 1,197
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
6
votes
2
answers
803
views
Conjecture about partitions of the powerset without the empty set
I would like to have some ideas about possibilities of proving or disproving the following conjecture:
For any partition $\mathcal{F}=\{\mathcal{A_1},\ldots,\mathcal{A_m} \}$ of the powerset without ...
- 990
5
votes
2
answers
600
views
Minimum number of transpositions to make two multiset permutations equal
I think this problem should have a known solution, but I wasn't able to find any reference.
Consider a multiset of size $n \cdot m$: it has $n$ elements, and all element multiplicities equal to $m$.
...
- 990
4
votes
1
answer
975
views
What is the minimum number of independent sets for a graph with fixed numbers of vertices and edges?
Fix integers $V$ and $E_{\text{max}}$, and consider graphs $G$ with $V$ vertices and at most $E_{\text{max}}$ edges. What is the best lower bound that one can give on the number of distinct ...
- 1,233
4
votes
1
answer
886
views
Existence of triangle-free graphs for regular graphs of degree at most n/2
It is known that for triangle-free graphs, if they are $d$-regular, then $2d\leq n$, where $n$ is the number of vertices. In words, the degree is less than or equal to half the number of vertices (...
- 43
4
votes
3
answers
780
views
Does an $(x, bx)$-biregular graph always contain a $x$-regular bipartite subgraph?
I guess a discrete-mathematics-related question is still welcome in MO since I was new to the community and learned from this amazing past post. The following claim is a simplified and abstract form ...
- 43
3
votes
1
answer
259
views
Davenport constant $D(S_5)=10$ or $11$?
I am working on computing the Davenport constant $D(G)$
of symmetric groups, which is the minimal number $d$
such that every sequence of $d$
elements, possibly with repetitions, is one-product, i.e. ...
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 ...
- 990
3
votes
1
answer
297
views
Do there exist at least two sets whose union gives the universe in a certain intersection-closed family of sets?
[Crossposted at math.stackexchange.]
Originally I posted a slightly more complicated version of this question. I decided to edit and put it in this simplified form, because I think that if we can ...
- 990
3
votes
2
answers
296
views
Choosing subsets to cover larger sets
Consider the set $S=\{1,2,\ldots, n\}$, and let $a<b<n$. What is the minimum number $f(a,b)$ such that there exist $f(a,b)$ subsets of $S$ of size $a$ for which any subset of $S$ of size $b$ ...
- 175
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
1
vote
1
answer
153
views
Schur complement and depermuting an algorithm for $\mathsf{determinant}\bmod2$
Let $$M=\begin{bmatrix}A&B\\C&D\end{bmatrix}$$ be a matrix in $\mathbb F_2^{n\times n}$ where $A\in\mathbb F_2$ and $D\in\mathbb F_2^{(n-1)\times(n-1)}$ are square.
Through the determinant ...
- 13.9k
1
vote
2
answers
386
views
Lower bound for the size of a family of sets
Consider a family $\mathcal{G} = \{ A_1,B_1,\ldots,B_m \}$ of $m+1$ non-empty finite distinct sets with the following property:
$$A_1 \cap B_k = \emptyset, 1 \le k \le m$$
Let $\mathcal{F} = \{A_1 \...
- 990
0
votes
1
answer
234
views
Minimum number of elements needed to represent a lattice with a union-closed family of sets
I know that it is possible to represent every finite lattice $L$ with a union-closed family $\mathcal{F}$ containing the empty set: for every $x\in L$, let $S_x=\{y\in L\, :\, y\not\geq x\}$ and $\...
- 990
0
votes
0
answers
104
views
Lower bound for couples of disjoint sets in some partitions of the power set
Originally posted on MathStackExchange but without answers.
Consider partitions $\mathcal{F}=\{\mathcal{A_1},\ldots,\mathcal{A_n} \}$ of the powerset without the empty set element $Q = \mathcal{P}([n])...
- 990
0
votes
0
answers
182
views
Another conjecture about couples of disjoint two-element sets
Let $\{A_1,B_1\},\ldots,\{A_k,B_k\}$ be all the distinct unordered couples of subsets, with $A_i \cap B_i = \emptyset, 1 \le i \le k$, that can formed from a set $\{C_1,\ldots,C_q\}$, $q \le \binom{n}{...
- 990