All Questions
12 questions
6
votes
1
answer
275
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
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
1
vote
1
answer
205
views
Chromatic number or independence number of the generalized Kneser Graph
For positive integers $n,k$ and $s$, where $0\le s<k$ and $k \le n$, we define the generalized Kneser graph $K(n,k,s)$ as follows: The vertices of $K(n,k,s)$ are the $k$-subsets of $[2n]$, i.e., we ...
- 135
4
votes
0
answers
176
views
Can resolution of the Kadison-Singer Problem provide progress on the Komlos Conjecture?
This is not a concrete question, just some thoughts.
The Komlos Conjecture is as follows-
There exists an absolute constant $C>0$, such that the following holds:
For all $d$ and any set of vectors ...
- 150
7
votes
0
answers
113
views
A question related to the union-closed sets conjecture
Let $f(n)$ denote the maximum possible cardinality of a collection $\mathcal F$ of nonempty sets which is closed under unions ($X,Y\in\mathcal F\implies X\cup Y\in\mathcal F$) and is such that no ...
- 13.4k
2
votes
1
answer
130
views
Distinguishing points by sets of given size
The problem is:
Given a finite set $X$ with size $x$ and let $B$ denote a family of $k$-element subsets of $X$, called blocks. What is the smallest possible number $n$ of blocks such that every ...
- 9,501
3
votes
0
answers
122
views
Generalization of fisher inequality
What upper bounds are known on the size of a family $\mathcal{S}$ of subsets $S_i \subset [N]$ such that:
i) each $S_i$ is of size $pk$.
ii) for $i \neq j$, $|S_i \cap S_j| \bmod p \in U$, for some ...
- 131
0
votes
1
answer
152
views
Erdős-Ko-Rado with intersections of size at least two
Up to how many subsets of $\{1,2,\dots,2n\}$ of size $n$ can we choose so that each pair has an intersection of size at least two?
The original Erdős-Ko-Rado paper shows that taking all subsets that ...
- 1,209
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
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
4
votes
2
answers
202
views
open question on intersecting rectangles - reference request
In the now-inactive (and very nice) "Tricki" website that describes various proof techniques, there is an article (written I think by Timothy Gowers) that describes the following Ramsey-type problem:
...
- 711
5
votes
4
answers
552
views
Better bounds for exact-intersection Erdős–Ko–Rado system?
What is the largest possible number of subsets of a $4n$-element set $X$, such that each subset contains precisely $2n$ elements, and such that each of the pairwise intersections of the subsets has ...
- 2,350