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Results tagged with co.combinatorics
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user 9924
Enumerative combinatorics, graph theory, order theory, posets, matroids, designs and other discrete structures. It also includes algebraic, analytic and probabilistic combinatorics.
11
votes
List of integers without any arithmetic progression of n terms
This is not a complete answer, but a reasonable estimate.
Fix a prime $(1-o(1))n<p<n$. Every length-$n$ arithmetic progression with the difference co-prime with $p$ contains an element divisible by $ …
- 22.6k
4
votes
Accepted
The definition of dissipative set
Maybe, you mean "a dissociated set"? A finite subset $A$ of an abelian group is called dissociated if all of its $2^{|A|}$ subset sums are pairwise distinct. It is certainly true that if $A$ is set of …
- 22.6k
37
votes
4
answers
2k
views
"Circular" domination in ${\mathbb R}^4$
The following problem is related to (and motivated by) the first open case of this MO question. It is difficult to believe that this is a hard problem; and yet, I do not have a solution.
For two vect …
- 22.6k
2
votes
Accepted
Minimum real number for subset sum difference
As I wrote in my comment, the trivial bounds are $1/2^{n-1}$ and $n/(2^n-1)$. Here is a proof of the estimate $b(n)<3\sqrt n/2^n$; maybe it can be improved further using the same idea.
Consider the …
- 22.6k
2
votes
Erdos-Szekeres Theorems
Firstly, I second Qiaochu's remark that I've never heard the Ramsey theorem referred to as Erdős-Szekeres' theorem.
Secondly, it is is true (and actually well-known) that the Ramsey theorem implies …
- 22.6k
5
votes
Accepted
Balanced vectors
It is a classical result of Barany and Grinberg (generalizing an earlier result of Spencer) that there exist $\lambda_1,\dotsc,\lambda_N\in\{\pm 1\}$ with
$$ \|\lambda_1a_1+\dotsb+\lambda_Na_N\| \le …
- 22.6k
3
votes
Accepted
Looking for a paper of Kemperman on semigroups
For all those interested, a scan of Kemperman's paper can be found here.
- 22.6k
2
votes
Accepted
Intersections of translates of finite sets of integers
For the case $t=2$, letting $A_1:=A$, $A_2:=-B$, $A_3:=[-k,k]$, and $A_4:=[-l,l]$, the sum in the left-hand side counts the number of quadruples $(a_1,a_2,a_3,a_4)\in A_1\times A_2\times A_3\times A_ …
- 22.6k
5
votes
Largest $A\subset \mathbb{F}_2^n$ such that no two $a\neq b$ in $A$ add to an element of $A.$
The condition $a\ne b$ can in fact be safely dropped: if $2\cdot A:=\{a+b\colon a,b\in A, a\ne b\}$ is disjoint from $A$, then $0\notin A$ (unless $A\subseteq\{0\}$) and therefore also the larger set …
- 22.6k
15
votes
Accepted
Multiples in sets of positive upper density
Not necessarily: you can in fact have $M_k(A)=\varnothing$ for all integer $k\ge 2$. This was shown by Besicovitch ("On the density of certain sequences of integers", Math. Ann. 110 (1935), no. 1, 336 …
- 22.6k
6
votes
Accepted
Infinite subset of $\mathbb{N}$ almost avoiding all "zebra crossings"
Take the $k$th element of $A$ to be $a_k:=2k!-1$, for all $k\ge 1$. If $k\ge n$, then $a_k\equiv -1\!\!\pmod{2n}$, whence $a_k\notin Z_n$. Thus, for any fixed $n$, there are only finitely many element …
- 22.6k
3
votes
Accepted
Smallest $k$ such that every vector is a linear combination of at most $k$ generators
A related problem was studied in the papers by Ben Klopsch and myself How long does it take to generate a group? and Generating abelian groups by addition only. Describing precisely the connections wi …
- 22.6k
6
votes
An optimization problem in finite groups
If $A,B\subseteq G$ satisfy $A^{-1}\cdot B=G$ and $|A|+|B|<K\sqrt{|G|}$ for some absolute constant $K$, then for the set $S:=A^{-1}\cup B$ we have $S\cdot S=G$ and $|S|<2K\sqrt{|G|}$. Thus, if you cou …
- 22.6k
6
votes
Covering a (hyper)cube with lines
Not an answer, but a comment too long to fit the space.
You may be interested to know that for finite projective geometries, the
property in question has a dedicated name: namely, a set $A\subset PG( …
- 22.6k
1
vote
Ways to "regularize" a graph
Here is a simple and "natural" (whatever it means) way to get a regular graph from a given graph $G$. Take two disjoint copies of $G$, and insert edges between all pairs of vertices $(v_1,v_2)$ (with …
- 22.6k