Questions tagged [infinite-combinatorics]
Combinatorial properties of infinite sets. This is a corner-point of set theory and combinatorics.
572 questions
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Chromatic number of the singleton-intersection graph on ${\cal P}(\omega)$
We endow $\newcommand{\Po}{{\mathcal P}(\omega)}\Po$ with a graph structure by letting
$E = \big\{\{a,b\}: a, b\subseteq \omega \land a \neq b \land (\exists n\in \omega(a\cap b = \{n\})\big\}$.
Every ...
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Infinite fibers in $K$-continuous maps $f:\mathbb{Z}\times \mathbb{Z} \to \mathbb{N}$
If $\newcommand{\Z}{\mathbb{Z}}\newcommand{\N}{\mathbb{N}}f:\Z\times\Z \to \N$ we let $${\rm I}(f)=\{n\in\N : f^{-1}(\{n\}) \text{ is infinite}\}$$ be the set of natural numbers with infinite pre-...
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Order-universal permutation numbers
Inspired by Dominic van der Zypen's question and Ilya Bogdanov's answer I wonder what to make of two cardinal characteristics. To define them, let $e_U$ be the increasing enumeration of an infinite ...
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Order-universal permutation $\alpha:\omega\to\omega$
For the set $\omega$ of non-negative integers, we let $\newcommand{\oo}{[\omega]^\omega}\oo$ be the collection of infinite subsets of $\omega$. If $U\in \oo$, there is a unique order-preserving ...
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Chains of nulls sets: on a question of Elkies (sort of)
Noam Elkies maintains a page of mathematical miscellany on his website. The last entry on this page is a problem he proposed to the American Mathematical Monthly, rejected on the advice of both ...
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Cardinal arithmetic inequalities according to ZF
Suppose $\kappa$, $\lambda$, $\mu$, and $\nu$ are cardinals which may or may not be ordinals. Can we prove without resorting to the axiom of choice either of the following:
$\kappa + \lambda \...
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Cardinality of fibers in a $T_2$-hypergraph with large edges
We say that a hypergraph $H=(V,E)$ is $T_2$ if for all $v, w\in V$ with $v\neq w$, there are disjoint sets $e_1, e_2\in E$ with $v\in e_1, w\in e_2$.
For $v\in V$, let $e_v= \{e\in E: v\in e\}$.
Given ...
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Chromatic number of the antichain hypergraph on $\mathcal P(\omega)$
If $H=(V, E)$ is a hypergraph, the its chromatic number $\chi(H)$ is the smallest non-empty cardinal $\kappa$ such that there is a map $c:V \to \kappa$ such that for every $e\in E$ containing more ...
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Ascending chains in $\mathcal{P}(\kappa)/I_{\mathrm{NS}}$
$\newcommand{\NS}{\mathrm{NS}}\mathcal{P}(\kappa)/I_{\NS}$ is the Boolean algebra of subsets of $\kappa$ under nonstationary symmetric difference, with $\mathbf{0} = [\emptyset] = I_{\NS}$, $\mathbf{1}...
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Chromatic number of the maximal chain hypergraph on ${\cal P}(\omega)$
If $H=(V, E)$ is a hypergraph, the its chromatic number $\chi(H)$ is the smallest non-empty cardinal $\kappa$ such that there is a map $c:V \to \kappa$ such that for every $e\in E$ containing more ...
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CH-preserving powerfully ccc forcing of size $2^{\aleph_1}$
Is it consistent with, or even implied by, CH that there is a CH-preserving, powerfully ccc complete Boolean algebra $\mathbb{B}$ of size $2^{\aleph_1}$? (Powerfully ccc means that ccc holds in every ...
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A proof by Woodin of the Kunen Theorem
Akihiro Kanamori in his book ''The higher infinity'', in the page 320 when presented the second proof of the Kunen inconsistency Theorem, states that there are a function $S:\kappa\rightarrow P(\...
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Relationship between tower number ($\mathfrak{t}$) and splitting number ($\mathfrak{s}$)
Recall that a tower is an infinite family $T$ of subsets of $\omega$ such that $T$ is well-ordered by the relation $\supset^*$ of almost inclusion and has no infinite pseudointersections. So, the ...
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Is $2^\kappa\rightarrow(\kappa^+)^2_2$ possible in $\mathsf{ZF}$?
It is a classical result of Sierpiński that $2^\kappa\not\rightarrow(\kappa^+)^2_2$ in $\mathsf{ZFC}$. The proof goes by picking a well-ordering $(2^\kappa,\prec)$ and comparing it with the ...
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Cardinal characteristic cofinalities
This question is motivated by a recent answer of mine to another question here. Generally I would like to know which problems regarding the cofinalities of cardinal characteristics are open, including,...
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Minimal separating subsets of $[\omega]^\omega$
Let $\newcommand{\om}{[\omega]^\omega}\om$ denote the collection of infinite subsets of the set of non-negative integers $\omega$. We say $A\subseteq\om$ is separating if for all $n,m\in\omega$ with $...
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Is there a way to visualize the Stone space of a Boolean algebra - i.e., via its corresponding lattice?
It is a little frustrating to me that I'm not able to 'see' clearly what the Stone space of a Boolean algebra is supposed to be - specially because we have a very visual object representing the ...
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Absolute derivative of a bijection $\varphi:\omega_+\to\omega_+$
Let $\newcommand{\ome}{\omega_+}\ome=\omega\setminus\{0\}$ denote the set of positive integers. For $a,b\in \ome$ we let $$||a-b|| = |(a\setminus b)\cup(b\setminus a)|$$ be the absolute difference of $...
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Why Catalan numbers appear
Let $(x_1,x_2,...,x_{2n+1 })$ be a sequence from 0 and 2, whose members satisfy all conditions
$$
\begin{align}
x_1&\le 1 \\
x_1&+x_2\le 2\\
\vdots &\qquad\vdots\\
x_1&+x_2+\ldots+x_{...
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Uniquely representing finite sets of integers by an intersection of infinite sets
$\def\N{\mathbb{N}}$Is there a set ${\cal C} \subseteq {\cal P}(\N)$ consisting of infinite subsets of $\N$, such that for every finite subset $A\subseteq \N$ there are two distinct and unique members ...
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What are the possible probabilities that a "close-to-uniform random permutation" of $\mathbb{N}$ will be a derangement?
This question builds upon a Math.SE post What is the probability that a geometric random permutation of $\mathbb{N}$ has no fixed points?. In this post $\mathbb{N}$ excludes $0$.
As discussed in a ...
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Infinite additive partition of $\mathbb{N}$
$\def\N{\mathbb{N}}$For any set $S\subseteq\N$ and $a\in \N$, we let $a+S =\{a+s: s\in S\}$. Are there infinite sets $A, S\subseteq \N$ such that $\{a+S: a\in A\}$ is an infinite partition of $\N$?
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Partitioning a cube into cuboids of different dimensions
This is how I have tried:
Initial stage: One triplet of the form $(n,n,n)$.
Second stage: Decompose original triplet into two triplets by splitting one of the elements of $(x,y,z)$ into two parts at ...
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Is the powerset graph construction injective?
$\def\Pow{\text{Pow}}$Let $G = (V,E)$ be a simple, undirected graph with $V\neq\varnothing $. We define the powerset graph of $G$ by the following:
the set of vertices is ${\cal P}(V)\setminus\{\...
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Can a Cohen real add a Kurepa tree?
If $V$ has no Kurepa tree and $G$ is $\mathrm{Add}(\omega,1)$-generic over $V$, can $V[G]$ have a Kurepa tree? More generally, can a forcing of size $\kappa$ create a $\kappa^+$-Kurepa tree?
This is ...
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How to show that the independence number $\mathfrak{i}(\kappa)$ is well-defined for some cardinal $\kappa$?
At the moment I am reading about cardinal characteristics of $\kappa$ for some uncountable $\kappa$. Here a lot of definitions can be generalised from the countable easily. For example for the pseudo-...
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Minimal full edge sets making a family of functions into graph homomorphisms
If $V$ is a set, we define $V^V$ to be the collection of all functions $f:V\to V$. If $(V_i, E_i)$ are simple, undirected graphs for $i=0,1$, that is $E\subseteq {V\choose 2}$, we say that a map $f:...
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Is the partition relation $\omega^\alpha\to(\omega^\alpha,\omega)$ true?
The partition relation $\beta\to(\gamma,\delta)$ means that for every function $f:[\beta]^2\to2$, either there is a subset $H$ of $\beta$ of order type $\gamma$ such that $f$ is $0$ on $[H]^2$, or ...
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Reverse Chang's conjecture
The two-cardinal transfer property $(\kappa,\lambda)\rightarrow(\kappa',\lambda')$ says "any structure of type $(\kappa,\lambda)$ has an elementarily equivalent structure of type $(\kappa',\...
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Minimal cardinality of families in $[\omega]^\omega$ dominating from below
For $A, B\subseteq \omega$ we write $A\subseteq^*B$ if $A\setminus B$
is finite. Let $[\omega]^\omega$ be the collection of infinite subsets.
We say that ${\cal D}\subseteq [\omega]^\omega$ is ...
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Failures of $\square$ from Chang-type reflection
For infinite cardinals $\nu \leq \lambda, \mu \leq \kappa$, let $\langle \kappa, \mu \rangle \twoheadrightarrow \langle \lambda, \nu \rangle$ be the assertion that, whenever $\langle f_i: i < \...
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Lengths of arithmetical sequences and arithmetical images for bijections $\varphi:\mathbb{N}\to\mathbb{N}$
We call a finite subset $S\subseteq \mathbb{N}$ arithmetical if there are $n, k\in\mathbb{N}$ with $k>1$ such that $S = \{n+j: 0 \leq j\leq k\}$.
Given an integer $\ell>0$ and a bijection $\...
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What are the uncountable homogenous graphs?
A graph is homogenous if every isomorphism between two induced finite subgraphs extends to an automorphism of the whole graph. All finite and countable homogenous graphs are known, forming a small ...
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For infinitely generated groups, is there a known relationship between its Specker ends and the combinatorial ends of its Cayley graphs?
Ends for groups are usually defined for finitely generated groups, where a lot of different definitions (Freudenthal's topological, combinatorial, metric, by covering spaces, etc) coincide. Specker ...
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Interchanging limits
The following definition is by Sinclair, G.E. A finitely additive generalization of the Fichtenholz–Lichtenstein theorem. Transactions of the American Mathematical Society. 1974;193:359-74.
A function ...
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Covering the positive integer point lattice in $\mathbb{R}^2$ with sublattices
Let $P$ be the positive integer point lattice on the plane, that is, all points $(x,y)\in\mathbb{R}^2$ such that $x,y\in\mathbb{N}, x,y>0$. Take $a_i,b_i,c_i\in P$ such that $b_i=(b_{i,x},b_{i,y}) \...
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Degrees in infinite graphs $G$ with $G\cong L(G)$
Motivation. For any graph $G$, let $L(G)$ denote its line graph. A graph $G$ with $G\cong L(G)$ is said to be line-graph (LG)-invariant. It turns out that the only finite connected LG-invariant graphs ...
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Generalized club of elementary submodels
This is a follow-up question to this previous question. A positive answer within ZFC would lead directly to a ZFC-proof of the
main result of this paper.
Let $ \kappa $ and $ \varTheta $ be ...
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Polynomially normal binary sequences
Motivation. When we try to construct a (pseudo-)random sequence $s:\newcommand{\N}{\mathbb{N}}\N\to\{0,1\}$ we often want $s$ itself, and some of its subsequences, to be normal.
Question. Is there a ...
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Do Suslin forests exist in $L$?
A $(\kappa,\lambda)$-forest is a collection $\mathcal{F}$ of functions such that:
for every $f\in\mathcal{F}$, we have $\mathrm{dom}(f)\in\mathcal{P}_\kappa(\lambda)$ and $\mathrm{ran}(f)\subseteq\{0,...
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Generalized clubs and clubs
I am looking for connections between Jech's version of the generalized club filter and the usual club filter. My motivation and ultimate goal is to turn this consistency result into a ZFC-proof.
Let $ ...
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Surjective hash functions $h:\{0,1\}^* \to \{0,1\}^{2n}$ with avalanche effect
Motivation. In computer science, hash functions are maps that convert binary strings of arbitrary length to a fixed-length binary string. In symbols, we have a map $h:\{0,1\}^* \to \{0,1\}^n$ for some ...
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Diamonds on supercompact $\kappa$ after a $\kappa$-c.c. forcing
Let $\kappa$ be supercompact. Then the (supercompact) Laver diamond holds at $\kappa$: There is $f:\kappa\to V_\kappa$ such that for all $\lambda\geq \kappa$ and $x\in H(\lambda^+)$ there is $j:V\to M$...
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Diamonds at $\omega_2$ under PFA
Let $\lambda$ be a cardinal, and $S\subset \lambda^+$ be stationary. A $\diamondsuit^+(S)$-sequence is a sequence $\langle \mathcal{A}_\alpha\mid \alpha\in S\rangle$ such that each $\mathcal{A}_\alpha\...
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Maximum density of sum-free sets with respect to Knuth's "addition"
A subset $S\subseteq\mathbb{N}$ is said to be sum-free if whenever $s,t\in S$, then $s+t\notin S$. For instance the set of odd numbers is sum-free and has (lower and upper) asymptotic density 1/2.
...
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Shifting an irrational binary sequence
Let $\newcommand{\tn}{\{0,1\}^\mathbb{N}}\tn$ be the collection of all infinite binary sequences. For $s\in\tn$ and $k\in\mathbb{N}$ let the left-shift of $s$ by $k$ positions, $\ell_k(s)\in \tn$, be ...
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Image and pre-image integer choice function
Let $\newcommand{\Nplus}{\mathbb{N}^+}\Nplus$ denote the set of positive integers. Is there a function $f:\Nplus\to\Nplus$ with the following property?
For all $(a,b)\in \Nplus\times\Nplus$ there is ...
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Is there an uncountable extension of the Ramsey set $[\omega]^2$?
We say that a family ${\cal A}\subseteq {\cal P}(\omega)$ is Ramsey
if for every map $c:{\cal A}\to\{0,1\}$ there is an infinite set $X\subseteq \omega$
with the following properties:
${\cal A}\cap {\...
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Is there a sparse almost disjoint family over $\omega$ of cardinality $2^{\aleph_0}$?
Is there an almost disjoint family $\mathcal{F}$ of subsets of $\omega$ of cardinality $2^{\aleph_0}$ satisfying the following property?
For all $A,B\in\mathcal{F}$ with $A\neq B$ and every $k\in\...
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Maximal Ramsey families
We say that a family $\mathcal R\subseteq \mathcal P(\omega)$ is Ramsey if
$\bigcup \mathcal R = \omega$, and
for every map $f:\mathcal R \to \{0,1\}$ there is an infinite set $X\subseteq \omega$
...
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