Related papers: Erd\H{o}s-Gy\'{a}rf\'{a}s problem for partially or…
We say that a poset $(Q,\le_{Q})$ contains an induced copy of a poset $(P,\le_P)$ if there is an injective function $\phi\colon P\to Q$ such that for every two $X,Y\in P$,\;\;$X\le_P Y$ if and only if $\phi(X)\le_Q \phi(Y)$. We denote the…
Given partially ordered sets (posets) $(P, \leq_P)$ and $(P', \leq_{P'})$, we say that $P'$ contains a copy of $P$ if for some injective function $f: P\rightarrow P'$ and for any $X, Y\in P$, $X\leq _P Y$ if and only of $f(X)\leq_{P'}…
Fix positive integers $p$ and $q$ with $2 \leq q \leq {p \choose 2}$. An edge-coloring of the complete graph $K_n$ is said to be a $(p, q)$-coloring if every $K_p$ receives at least $q$ different colors. The function $f(n, p, q)$ is the…
In this thesis, we present quantitative Ramsey-type results in the setting of finite sets that are equipped with a partial order, so-called posets. A prominent example of a poset is the Boolean lattice $Q_n$, which consists of all subsets…
A $(p,q)$-coloring of a graph $G$ is an edge-coloring of $G$ which assigns at least $q$ colors to each $p$-clique. The problem of determining the minimum number of colors, $f(n,p,q)$, needed to give a $(p,q)$-coloring of the complete graph…
The Erd\H{o}s-Gy\'arf\'as number $f(n, p, q)$ is the smallest number of colors needed to color the edges of the complete graph $K_n$ so that all of its $p$-clique spans at least $q$ colors. In this paper we improve the best known upper…
A Boolean lattice $\mathcal{B}_n=(2^X, \leq)$ is the power set of an $n$-element ground set $X$ equipped with inclusion relation. For two posets $\mathcal{P}$ and $\mathcal{Q}$, we say that $\mathcal{Q}$ contains an \emph{induced copy} of…
A subposet $Q'$ of a poset $Q$ is a copy of a poset $P$ if there is a bijection $f$ between elements of $P$ and $Q'$ such that $x\leq y$ in $P$ iff $f(x)\leq f(y)$ in $Q'$. For posets $P, P'$, let the poset Ramsey number $R(P,P')$ be the…
We address the following rainbow Ramsey problem: For posets $P,Q$ what is the smallest number $n$ such that any coloring of the elements of the Boolean lattice $B_n$ either admits a monochromatic copy of $P$ or a rainbow copy of $Q$. We…
Motivated by the paper of Axenovich and Walzer [2], we study the Ramsey-type problems on the Boolean lattices. Given posets $P$ and $Q$, we look for the smallest Boolean lattice $\mathcal{B}_N$ such that any coloring on elements of…
The generalized Ramsey number $f(n, p, q)$ is the smallest number of colors needed to color the edges of the complete graph $K_n$ so that every $p$-clique spans at least $q$ colors. Erd\H{o}s and Gy\'arf\'as showed that $f(n, p, q)$ grows…
For two posets $(P,\le_P)$ and $(P',\le_{P'})$, we say that $P'$ contains a copy of $P$ if there exists an injective function $f\colon P'\to P$ such that for every two $X,Y\in P$, $X\le_P Y$ if and only if $f(X)\le_{P'} f(Y)$. Given two…
Given partially ordered sets (posets) $(P, \leq_P)$ and $(P', \leq_{P'})$, we say that $P'$ contains a copy of $P$ if for some injective function $f\colon P\rightarrow P'$ and for any $A, B\in P$, $A\leq _P B$ if and only if $f(A)\leq_{P'}…
For fixed $p$ and $q$, an edge-coloring of the complete graph $K_n$ is said to be a $(p, q)$-coloring if every $K_p$ receives at least $q$ distinct colors. The function $f(n, p, q)$ is the minimum number of colors needed for $K_n$ to have a…
Let $F(n,k)$ ($f(n,k)$) denote the maximum possible size of the smallest color class in a (partial) $k$-coloring of the Boolean lattice $B_n$ that does not admit a rainbow antichain of size $k$. The value of $F(n,3)$ and $f(n,2)$ has been…
A poset $(Q,\le_Q)$ contains an induced copy of a poset $(P,\le_P)$ if there exists an injective mapping $\phi\colon P\to Q$ such that for any two elements $X,Y\in P$, $X\le_P Y$ if and only if $\phi(X)\le_Q \phi(Y)$. By $Q_n$ we denote the…
We say that a poset $Q$ contains a copy (resp.~an induced copy) of a poset $P$ if there is an injection $f : P \to Q$ such that for any $x,y \in P$, $f(x)\leq f(y)$ in $Q$ if (resp.~if and only if) $x\leq y$ in $P$. Let $\mathcal{Q}=\{Q_{n}…
Given positive integers $p$ and $q$, a $(p,q)$-coloring of the complete graph $K_n$ is an edge-coloring in which every $p$-clique receives at least $q$ colors. Erd\H{o}s and Shelah posed the question of determining $f(n,p,q)$, the minimum…
Given positive integers $p,q$. For any integer $k\ge2$, an edge coloring of the complete $k$-graph $K_n^{(k)}$ is said to be a $(p,q)$-coloring if every copy of $K_p^{(k)}$ receives at least $q$ colors. The Erd\H{o}s-Gy\'{a}rf\'{a}s…
A subposet $Q'$ of a poset $Q$ is a \textit{copy of a poset} $P$ if there is a bijection $f$ between elements of $P$ and $Q'$ such that $x \le y$ in $P$ iff $f(x) \le f(y)$ in $Q'$. For posets $P, P'$, let the \textit{poset Ramsey number}…