Related papers: Boolean lattice without small rainbow subposets

We develop a powerful tool for embedding any tree poset $P$ of height $k$ in the Boolean lattice which allows us to solve several open problems in the area. We show that: * If $H$ is a family in $B_n$ with $|H|\ge (q-1+\varepsilon){n\choose…

Given graphs $G$ and $H$ and a positive integer $k$, the \emph{Gallai-Ramsey number}, denoted by $gr_{k}(G : H)$ is defined to be the minimum integer $n$ such that every coloring of $K_{n}$ using at most $k$ colors will contain either a…

Given a poset $P$, a family $F$ of elements in the Boolean lattice is said to be $P$-saturated if (1) $F$ contains no copy of $P$ as a subposet and (2) every proper superset of $F$ contains a copy of $P$ as a subposet. The maximum size of a…

Given a nonempty graph $G$, a collection of nonempty graphs $\cal{H}$, and a positive integer $k$, the Gallai-Ramsey number $\mathrm{gr}_k(G:\mathcal{H})$ is defined to be the minimum positive integer $n$ such that every exact…

For every integer $n$ with $n \geq 6$, we prove that the Boolean dimension of a poset consisting of all the subsets of $\{1,\dots,n\}$ equipped with the inclusion relation is strictly less than $n$.

Given two non-empty graphs $G,H$ and a positive integer $k$, the Gallai-Ramsey number $\operatorname{gr}_k(G:H)$ is defined as the minimum integer $N$ such that for all $n\geq N$, every $k$-edge-coloring of $K_n$ contains either a rainbow…

A colored complete graph is said to be Gallai-colored if it contains no rainbow triangle. This property has been shown to be equivalent to the existence of a partition of the vertices (of every induced subgraph) in which at most two colors…

The $k$-edge-colored bipartite Gallai-Ramsey number $\operatorname{bgr}_k(G:H)$ is defined as the minimum integer $n$ such that $n^2\geq k$ and for every $N\geq n$, every edge-coloring (using all $k$ colors) of complete bipartite graph…

Given two graphs $G$ and $H$, the \emph{general $k$-colored Gallai-Ramsey number} $\operatorname{gr}_k(G:H)$ is defined to be the minimum integer $m$ such that every $k$-coloring of the complete graph on $m$ vertices contains either a…

For a family $\mathcal{F}$ of subsets of [n]=\{1, 2, ..., n} ordered by inclusion, and a partially ordered set P, we say that $\mathcal{F}$ is P-free if it does not contain a subposet isomorphic to P. Let $ex(n, P)$ be the largest size of a…

The Boolean lattice $2^{[n]}$ is the power set of $[n]$ ordered by inclusion. A chain $c_{0}\subset...\subset c_{k}$ in $2^{[n]}$ is rank-symmetric, if $|c_{i}|+|c_{k-i}|=n$ for $i=0,...,k$; and it is symmetric, if $|c_{i}|=(n-k)/2+i$. We…

Given a finite poset $\mathcal{P}$, a family $\mathcal{F}$ of elements in the Boolean lattice is induced-$\mathcal{P}$-saturated if $\mathcal{F}$ contains no copy of $\mathcal{P}$ as an induced subposet but every proper superset of…

A family $\mathcal{G}$ of sets is a weak copy of a poset $P$ if there is a bijection $f:P\rightarrow \mathcal{G}$ such that $p\leqslant q$ implies $f(p)\subseteq f(q)$. If $f$ satisfies $p\leqslant q$ if and only if $f(p)\subseteq f(q)$,…

The problem of determining the maximum size $La(n,P)$ that a $P$-free subposet of the Boolean lattice $B_n$ can have, attracted the attention of many researchers, but little is known about the induced version of these problems. In this…

For two posets $P$ and $Q$, we say $Q$ is $P$-free if there does not exist any order-preserving injection from $P$ to $Q$. The speical case for $Q$ being the Boolean lattice $B_n$ is well-studied, and the optiamal value is denoted as…

Given a graph $G$ and a positive integer $k$, the \emph{Gallai-Ramsey number} is defined to be the minimum number of vertices $n$ such that any $k$-edge coloring of $K_n$ contains either a rainbow (all different colored) triangle or a…

Let $G, H$ be two non-empty graphs and $k$ be a positive integer. The Gallai-Ramsey number $\operatorname{gr}_k(G:H)$ is defined as the minimum positive integer $N$ such that for all $n\geq N$, every $k$-edge-coloring of $K_n$ contains…

A relational structure $\mathrm{R}$ is {\em rainbow Ramsey} if for every finite induced substructure $\mathrm{C}$ of $\mathrm{R}$ and every colouring of the copies of $\mathrm{C}$ with countably many colours, such that each colour is used…

A model named `Colored Percolation' has been introduced with its infinite number of versions in two dimensions. The sites of a regular lattice are randomly occupied with probability $p$ and are then colored by one of the $n$ distinct colors…

For two graphs $G,H$, the \emph{Ramsey number} $r(G,H)$ is the minimum integer $n$ such that any red/blue edge-coloring of $K_n$ contains either a red copy of $G$ or a blue copy of $H$. For two graphs $G,H$, the \emph{Gallai-Ramsey number}…