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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.
25
votes
2
answers
1k
views
The number of polynomials on a finite group
A function $f:X\to X$ on a group $X$ is called a polynomial if there exist $n\in\mathbb N=\{1,2,3,\dots\}$ and elements $a_0,a_1,\dots,a_n\in X$ such that $f(x)=a_0xa_1x\cdots xa_n$ for all $x\in X$. …
18
votes
Dividing a cake between $n-1$, $n$, or $n+1$ guests
Writing down the details of the argument of Ilya Bogdanov, we can obtain the following upper bound:
Theorem. $f(n)\le\frac83n-1$ for every $n\ge 2$.
Proof. If $n=3k+1$ or $n=3k+2$, then following th …
15
votes
1
answer
416
views
What is the smallest cardinality of a self-linked set in a finite cyclic group?
A subset $A$ of a group $G$ is defined to be self-linked if $A\cap gA\ne\emptyset$ for all $g\in G$. This happens if and only if $AA^{-1}=G$.
For a finite group $G$ denote by $sl(G)$ the smallest car …
11
votes
1
answer
388
views
Does every finite affine plane have the doubling property?
Definition 1. An affine plane is a pair $(X,\mathcal L)$ consisting of a set $X$ and a family $\mathcal L$ of subsets of $X$ called lines which satisfy the following axioms:
Any distinct points $x,y\ …
10
votes
1
answer
350
views
Is the group of translations of an affine plane always commutative?
$\DeclareMathOperator\Dil{Dil}\DeclareMathOperator\Trans{Trans}\DeclareMathOperator\Col{Col}$An affine plane is a set of points $X$ endowed with a family $\mathcal L$ of subsets of $X$, called lines, …
9
votes
0
answers
462
views
Measuring the randomness of texts
The question concerns statistic properties of random words in a finite alphabet $A$.
By $A^{<\omega}$ we denote the set of all words in the alphabet $A$, i.e. finite sequences of elements of $A$.
Let …
9
votes
1
answer
628
views
A reference to infinite version of the Sunflower Lemma
Please help me to find a proper reference to the following infinite version of the Sunflower Lemma.
Lemma. Let $n\in\mathbb N$. Every infinite family of $n$-element sets contains an infinite subfa …
8
votes
1
answer
358
views
A combinatorial property of uncountable groups, II
Problem 1. Is it true that each uncountable group $G$ contains two subsets $A,B\subset G$ such that
1) for any $x,y\in G$ the intersection $xA\cap yB$ is finite and
2) for any function $ …
7
votes
1
answer
325
views
Large gaps in Singer's difference sets
This question is related to the question I asked earlier.
For a natural number $n$, a set $D$ of integer numbers is called a $n$-cyclic difference set if each integer number $x\notin n\mathbb Z$ can …
6
votes
0
answers
190
views
The highest degree of a polynomial on a finite group
This question is motivated by the comments and the answer to this MO-question.
First let us recall some definitions.
A function $f:X\to X$ on a group $X$ is called a polynomial if there exists $n\in\m …
6
votes
1
answer
538
views
Does Playfair imply Proclus?
I am interested in the interplay between the Playfair and Proclus Axioms in linear spaces.
By a linear space I understand a pair $(X,\mathcal L)$ consisting of a set $X$ and a family $\mathcal L$ of s …
6
votes
1
answer
384
views
What is the cardinality of liners of rank 4? Is it always equal 27?
Definition 1. A binary operation $\cdot:X\times X\to X$, $\cdot:(xy)\mapsto xy$, on a set $X$ will be called a line operation if
$$xx=x,\quad xy=yx,\quad (xy)x=y$$
for every $x,y\in X$.
Remark 1. Ever …
5
votes
1
answer
357
views
The number of polynomials on a finite group, II
This question is follow up of this MO-post.
First let us recall the necessary definitions.
A function $f:X\to X$ on a group $X$ is called a polynomial if there exists $n\in\mathbb N$ and elements $a_0 …
5
votes
Accepted
Minimal covers in hypergraphs with finite edges
Let $V:=\omega\times\omega$ and $E=\{E_{n,m}:n,m\in\omega\}$ where $$E_{n,m}:=(\{0,\dots,n\}\times\{m\})\cup\{(2n,m+1)\}.$$ It seems that the hypergraph $(V,E)$ has no minimal cover.
A simplification …
5
votes
5
answers
562
views
Is every uniform hyperbolic linear space infinite?
I start with definitions.
Definition 1. A linear space is a pair $(X,\mathcal L)$ consisting of a set $X$ and a family $\mathcal L$ of subsets of $X$ satisfying three axioms:
(L1) for any distinct poi …