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Problem. Is there an infinite set $I\subset 3^\omega$ such that for any infinite subset $J\subset I$ there exists $n\in\omega$ such that $\{x(n):x\in J\}=3$? Here $3^\omega$ is the set of functions f …

In a wider context, there is a well-known list of 17 formulas (selected by Ian Stewart) that changed the course of history, see https://www.businessinsider.com/17-equations-that-changed-the-world-2014 …

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 $x\in X$. …

In this preprint (written jointly with Alex Ravsky) we prove the following partial answers to this problem. First some definitions. A non-empty subset $D$ of an Abelian group is called decomposable if …

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 …

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 …

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 …

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 …

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 …

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 …

The answer to this question is "No". A non-affine biaffine Cartesian plane can be constructed as follows. First, we fix a suitable terminology. Every function $F:X\to Y$ is identified with its graph …

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 …

This question concerns the (synthetic) geometry of linear spaces. Definition 1. A linear space is a pair $(P,\mathcal L)$ consisting of a set $P$ whose elements are called points and a family $\mathca …

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 $ …

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 …