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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 …

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 …

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 …

Suppose you are assigning values in $S$ (assume $|S|<\infty$) to nodes of a (directed) graph in a stochastic way. At the beginning, none of the node is assigned values. At the $i^{th}$ step, you (unif …

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 …

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 $0\leq k\leq 3$ …

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 …

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 …