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Suppose that ${\cal C}$ is a set of subsets of $\{1,\ldots,n\}$ with the following properties: $\{1,\ldots,n\}\notin {\cal C}$, for all $x,y\in \{1,\ldots, n\}$ there is $A\in {\cal C}$ such that $\ …

Let $X\neq \emptyset $ be a finite set and suppose that ${\cal C}$ is a set of subsets of $X$ with the following properties: $X\notin {\cal C}$, and for all $x,y\in X$ there is $A\in {\cal C}$ such …

Is Rota's basis conjecture implied by the same statement restricted to vector spaces over a fixed field (say $\mathbb{R}$, or even $\mathbb{Z}_2$)?

(This is a follow-up to this question.) Let $X\neq \emptyset$ be a finite set and suppose that ${\cal C}$ is a set of subsets of $X$ with the following properties: all members of ${\cal C}$ contain …

Informal version What is the maximum number of distinct $n$-runs that a $\{0,1\}$-sequence of length $2^n$ can have? Formal version If $A, B$ are sets, we denote by $B^A$ the set of all functions $f:A …

This question is motivated by Frankl's union-closet sets conjecture. Let $n\in\mathbb{N}$ and set $[n] = \{0,1,\ldots,n\}$. We say that a family ${\cal A} \subseteq {\cal P}([n])$ is union-closed if …

By a graph I mean a pair $G = (V, E)$ where $V$ is a set and $E \subseteq [V]^2 := \{\{a,b\}: a\neq b \in V\}$. A graph homomorphism between graphs $G, H$ is a map $f:V(G)\to V(H)$ such that $\{v, w\} …

For any set $X$ we set $[X]^2 = \big\{\{a,b\}: a, b\in X\text{ and } a\neq b\big\}$. We say a simple undirected graph $G=(V,E)$ is an $n$-clique graph if there are $S_1,\ldots,S_n\subseteq V$ such th …

Let $G=(V,E)$ be a simple, undirected graph. We call a partition ${\cal P}$ of a non-empty subset of $V$ a Hadwiger partition if every block (member of ${\cal P}$) is non-empty and connected, and i …

Is there a set $X\neq\emptyset$ and a collection ${\cal F}\subseteq {\cal P}(X)\setminus\{\emptyset\}$ of non-empty subsets of $X$ with the following properties? $a\in {\cal F} \implies |a|\geq 2$, …

(A version of this question for undirected graphs.) Let $G=(V,E)$ be a finite, simple, undirected graph. For $v\in V$ set $$ N(v) := \{x\in V: \{x,v\}\in E\}. $$ Is it possible to find a partitio …

A covering of a non-empty set $X$ is a collection ${\cal U} \subseteq ({\cal P}(X)\setminus\{\emptyset\})$ such that $\bigcup {\cal U} = X$. If ${\cal U}$ is a covering of $X$ then a function $f:{\cal …

A covering of a non-empty set $X$ is a collection ${\cal U} \subseteq ({\cal P}(X)\setminus\{\emptyset\})$ such that $\bigcup {\cal U} = X$. If ${\cal U}$ is a covering of $X$ then a function $f:{\cal …

We call a map $f:{\mathbb Z}\times {\mathbb Z} \to {\mathbb Z}$ an additive grid if for all $x,y \in {\mathbb Z}$ we have that $f(x,y)$ is the sum of the neighboring values, that is, $$f(x,y) = f(x-1, …

For any positive integer $n$, let $[n] = \{1,\ldots,n\}$. Let $[n]^{[n]}$ denote the set of functions $f:[n]\to [n]$. For $f\in[n]^{[n]}$, we define the maximum accumulation $\text{macc}(f)$ by $$\tex …