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Enumerative combinatorics, graph theory, order theory, posets, matroids, designs and other discrete structures. It also includes algebraic, analytic and probabilistic combinatorics.
5
votes
1
answer
153
views
Ramseyan property of structure
A binary relation $R$ on $\mathbb{N}$ is $(n,k)$-Ramseyan iff for every coloring $f:[\mathbb{N}]^n\rightarrow l$, there exists a subset $G$ of $\mathbb{N}$
such that $(G,R)$ is isomorphic to $(\mathbb …
2
votes
0
answers
80
views
Small set in partition-large class
A collection $\mathcal{A}\subseteq \mathcal{P}(X)$
is $k$-large in $X$
if for every $k$-partition
of $X$ namely
$X_1,\cdots,X_k$, there exists an $i\leq k$ such that $X_i\in \mathcal{A}$;
$\mathcal …
3
votes
1
answer
141
views
The size of monochromatic submatrix
We say a matrix $(a_{ij})$ is 0-1 matrix if $a_{ij}\in \{0,1\}$ for all $i,j$. We say a matrix $(a_{ij})$ is monochromatic if for some $a$, $a_{ij} = a$ for all $i,j$.
Question: Let $c\geq 1/2$ be a …
1
vote
0
answers
138
views
A generalization of Hales-Jewett theorem
Hales-Jewett theorem ($HJ(\alpha,k,n)$) states that for every coloring $f:\alpha^N\rightarrow k$ where $N$ is sufficiently large, there is an $n$-dimensional combinatorial subspace of $\alpha^N$ that …
2
votes
1
answer
165
views
Covering subset with large probability
Let $c>0$, $0<\lambda<1$, and let $k\in \mathbb{N}$ be sufficiently large. Let $X$ be a uniformly random subset of $\{1,\cdots,N\}$. Denote by $[N]^x$ the collection of $[x]$-element subset of $\{1,\c …
4
votes
1
answer
90
views
Ramsey style theorem with unbounded colors
Question: Let $\varepsilon>0$ and $N\in\omega$ be sufficiently large (depending on $\varepsilon$).
Let $h:\subseteq N\rightarrow N$ be such that $h(B)\notin B$ for all $B\subsetneq N$.
Must there be …
2
votes
0
answers
59
views
Does periodic pattern arise in syndetic pattern
We wonder for two "large" sets $I,J\subseteq \omega$, if $(J-J)\cap I=\emptyset$, then it must be due to certain periodic pattern.
We say $I\subseteq \omega$ is $t$-syndetic iff for every $n\in\omega$ …
1
vote
0
answers
85
views
Winning criterion for a combinatorial game
Given $n$,
let $\mathcal{R}$ be a set of pairs $(\rho,A)$
where $A\subseteq n, \rho\in 2^A$.
Consider the following game between A and B.
At each round $t$, A enumerates an $m\in n$ (that has not been …
0
votes
1
answer
184
views
Combinatorially defined effectively closed set
Is there a combinatorially defined, nonempty effectively closed set $Q\subseteq 2^\omega$ such that all members of $Q$ are incomputable?
Combinatorially defined means that the definition of $Q$ does …
2
votes
1
answer
288
views
Is the consecutive sum set large in general?
$\DeclareMathOperator\CSS{CSS}$It is well known that for a set $A$ of integers, if $\gcd(A) = d$,
then the set of (integer) linear combinations of $A$ is $d\mathbb{Z}$.
I'm looking for a probability g …
0
votes
1
answer
150
views
Finite Hindman theorem
Consider the following finite version Hindman theorem:
For every sufficiently large $N\in\omega$ and 2-partition of $N=N_0\cup N_1$, there are $i<2,a,b,c\in N_i$ such that $a+b=c$.
The only proof I kn …
6
votes
1
answer
349
views
Ramsey type theorem
Let $\mathcal{P}(\{0,\dotsc,7\})$ denote the power set of $\{0,\dotsc,7\}$.
Is the following true?
For any function $f: \mathcal{P}(\{0,\dotsc,7\})\rightarrow\{0,1\}$ there exists $0\leq k\leq 3$ …
3
votes
1
answer
222
views
Density of a somewhat random set
The density of a set
$X\subseteq\omega$ refers to:
$\limsup\limits_{n\rightarrow\infty}\dfrac{C\cap n}{n}$.
Given a set of positive integers
$F= \{m_0<\cdots<m_{k-1}\}$,
let $C\subseteq \omega$ be s …
4
votes
0
answers
111
views
Set version of ramsey type problem
For two sets of numbers $A,B$, write $A<B$ iff $\max A<\min B$.
For a sequence of integers $a_0,\cdots,a_{n-1}>0$,
let $Prop(a_0,\cdots,a_{n-1})$ denote the following proposition:
Given $n$ sets of i …
12
votes
2
answers
1k
views
Graph automorphism group
Let $A_w$ denote such set of positive integer $n$ that: for any two permutations $\pi_0,\pi_1\in S_n$, if $\pi_1$ is not a power of $\pi_0$, then there exists a (labeled non oriented) graph $G$ of siz …