相关论文: The Anti-Ramsey Problem for the Sidon equation

In this paper, we first study a new extremal problem recently posed by Conlon and Tyomkyn~(arXiv: 2002.00921). Given a graph $H$ and an integer $k\geqslant 2$, let $f_{k}(n,H)$ be the smallest number of colors $c$ such that there exists a…

For a graph $L$ and an integer $k\geq 2$, $R_k(L)$ denotes the smallest integer $N$ for which for any edge-colouring of the complete graph $K_N$ by $k$ colours there exists a colour $i$ for which the corresponding colour class contains $L$…

Consider families of $k$-subsets (or blocks) on a ground set of size $v$. Recall that if all $t$-subsets occur with the same frequency $\lambda$, one obtains a $t$-design with index $\lambda$. On the other hand, if all $t$-subsets occur…

Given a graph $G$ and a subgraph $H$ of $G$, let $rb(G,H)$ be the minimum number $r$ for which any edge-coloring of $G$ with $r$ colors has a rainbow subgraph $H$. The number $rb(G,H)$ is called the rainbow number of $H$ with respect to…

We find explicitly all rational solutions of the title equation for all integers $k$ in the range $|k|\leq 50$ except for $k=-47,-39$. For the solution, a variety of methods is applied, which, depending on $k$, may range from elementary,…

Consider the set $[m]\times [n] = \{(i,j)\, : 1\le i \le m, 1\le j \le n\}$ and the equation $x_1+x_2 = x_3$, namely $eq$. The \emph{rainbow number of $[m] \times [n]$ for $eq$}, denoted $\text{rb}([m]\times [n],eq)$, is the smallest number…

Considering a natural generalization of the Ruzsa-Szemer\'edi problem, we prove that for any fixed positive integers $r,s$ with $r<s$, there are graphs on $n$ vertices containing $n^{r}e^{-O(\sqrt{\log{n}})}=n^{r-o(1)}$ copies of $K_s$ such…

In an edge-colored graph $G$, a rainbow clique $K_k$ is a $k$-complete subgraph in which all the edges have distinct colors. Let $e(G)$ and $c(G)$ be the number of edges and colors in $G$, respectively. In this paper, we show that for any…

Let the integers $1,\ldots,n$ be assigned colors. Szemer\'edi's theorem implies that if there is a dense color class then there is an arithmetic progression of length three in that color. We study the conditions on the color classes forcing…

We consider following geometric Ramsey problem: find the least dimension $n$ such that for any 2-coloring of edges of complete graph on the points $\{\pm 1\}^n$ there exists 4-vertex coplanar monochromatic clique. Problem was first analyzed…

We say that an edge-coloring of a graph $G$ is proper if every pair of incident edges receive distinct colors, and is rainbow if no two edges of $G$ receive the same color. Furthermore, given a fixed graph $F$, we say that $G$ is rainbow…

The rainbow Tur\'an number, a natural extension of the well studied traditional Tur\'an number, was introduced in 2007 by Keevash, Mubayi, Sudakov and Verstra\"ete. The rainbow Tur\'an number of a graph $H$, $ex^{*}(n,H)$, is the largest…

A family $\mathcal{G}$ of sets is a copy of a poset $(P,\leqslant)$ if $(\mathcal{G},\subseteq)$ is isomorphic to $(P,\leqslant)$. The forbidden subposet problem asks for determining $La^*(n,P)$, the maximum size of a family…

In a properly edge colored graph, a subgraph using every color at most once is called rainbow. In this thesis, we study rainbow cycles and paths in proper edge colorings of complete graphs, and we prove that in every proper edge coloring of…

For $t \in \mathbb{N}$, we say that a colouring of $E(K_n)$ is $\textit{almost}$ $t$-$\textit{Gallai}$ if no two rainbow $t$-cliques share an edge. Motivated by a lemma of Berkowitz on bounding the modulus of the characteristic function of…

For a fixed family of $r$-uniform hypergraphs $\mathcal{F}$, the anti-Ramsey number of $\mathcal{F}$, denoted by $ ar(n,r,\mathcal{F})$, is the minimum number $c$ of colors such that for any edge-coloring of the complete $r$-uniform…

Schur's Theorem states that, for any $r \in \mathbb{Z}^+$, there exists a minimum integer $S(r)$ such that every $r$-coloring of $\{1,2,\dots,S(r)\}$ admits a monochromatic solution to $x+y=z$. Recently, Budden determined the related…

For positive integers $n,r,s$ with $r > s$, the set-coloring Ramsey number $R(n;r,s)$ is the minimum $N$ such that if every edge of the complete graph $K_N$ receives a set of $s$ colors from a palette of $r$ colors, then there is guaranteed…

In this paper we present three different results dealing with the number of $(\leq k)$-facets of a set of points: 1. We give structural properties of sets in the plane that achieve the optimal lower bound $3\binom{k+2}{2}$ of $(\leq…

We provide several constructions for problems in Ramsey theory. First, we prove a superexponential lower bound for the classical 4-uniform Ramsey number $r_4(5,n)$, and the same for the iterated $(k-4)$-fold logarithm of the $k$-uniform…