Related papers: A sparse canonical van der Waerden theorem
A hypergraph $H$ is properly colored if for every vertex $v\in V(H)$, all the edges incident to $v$ have distinct colors. In this paper, we show that if $H_{1}$, \cdots, $H_{s}$ are properly-colored $k$-uniform hypergraphs on $n$ vertices,…
We compare the strength of polychromatic and monochromatic Ramsey theory in several set-theoretic domains. We show that the rainbow Ramsey theorem does not follow from ZF, nor does the rainbow Ramsey theorem imply Ramsey's theorem over ZF.…
A path in an edge-colored graph $G$ is called a rainbow path if no two edges of the path are colored the same. The minimum number of colors required to color the edges of $G$ such that every pair of vertices are connected by at least $k$…
The size-Ramsey number $\hat{R}(F)$ of a graph $F$ is the smallest integer $m$ such that there exists a graph $G$ on $m$ edges with the property that any colouring of the edges of $G$ with two colours yields a monochromatic copy of $F$. In…
A subgraph of an edge-coloured graph is called rainbow if all its edges have distinct colours. The study of rainbow subgraphs goes back to the work of Euler on Latin squares and has been the focus of extensive research ever since. Many…
A perfect matching M in an edge-colored complete bipartite graph K_{n,n} is rainbow if no pair of edges in M have the same color. We obtain asymptotic enumeration results for the number of rainbow matchings in terms of the maximum number of…
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…
Every positive integer greater than a positive integer $r$ can be written as an integer that is the sum of powers of $r$. Here we use this to prove the conjecture posed by Ronald Graham, B. Rothschild and Joel Spencer back in the nineteen…
We prove that any continuous map of an N-dimensional simplex Delta_N with colored vertices to a d-dimensional manifold M must map r points from disjoint rainbow faces of Delta_N to the same point in M: For this we have to assume that N \geq…
Consider a homogeneous polynomial $p(z_1,...,z_n)$ of degree $n$ in $n$ complex variables . Assume that this polynomial satisfies the property : \\ $|p(z_1,...,z_n)| \geq \prod_{1 \leq i \leq n} Re(z_i)$ on the domain $\{(z_1,...,z_n) :…
The colored Tverberg theorem asserts that for every d and r there exists t=t(d,r) such that for every set C in R^d of cardinality (d+1)t, partitioned into t-point subsets C_1,C_2,...,C_{d+1} (which we think of as color classes; e.g., the…
If we want to color $1,2,\ldots,n$ with the property that all 3-term arithmetic progressions are rainbow (that is, their elements receive 3 distinct colors), then, obviously, we need to use at least $n/2$ colors. Surprisingly, much fewer…
For graphs $G$ and $H$, let $G {\displaystyle\smash{\begin{subarray}{c} \hbox{$\tiny\rm rb$} \\ \longrightarrow \\ \hbox{$\tiny\rm p$} \end{subarray}}}H$ denote the property that for every proper edge-colouring of $G$ there is a rainbow $H$…
We show that the threshold for having a rainbow copy of a power of a Hamilton cycle in a randomly edge colored copy of $G_{n,p}$ is within a constant factor of the uncolored threshold. Our proof requires $(1+\varepsilon)$ times the minimum…
An edge-colored graph is called \textit{rainbow graph} if all the colors on its edges are distinct. Given a positive integer $n$ and a graph $G$, the \textit{anti-Ramsey number} $ar(n,G)$ is defined to be the minimum number of colors $r$…
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…
An $r$-uniform hypergraph ($r$-graph for short) is called linear if every pair of vertices belong to at most one edge. A linear $r$-graph is complete if every pair of vertices are in exactly one edge. The famous Brown-Erd\H{o}s-S\'os…
We consider the chromatic number of the random Borsuk graph. The random Borsuk graph is obtained by sampling $n$ points i.i.d. uniformly at random on the $d$-dimensional sphere $S^d$, and joining a pair of points by an edge whenever their…
The Lov\'asz Local Lemma is a powerful probabilistic technique for proving the existence of combinatorial objects. It is especially useful for colouring graphs and hypergraphs with bounded maximum degree. This paper presents a general…
The work deals with the threshold probablity for r-colorability in the binomial model H(n,k,p) of a random k-uniform hypergraph. We prove a lower bound for this threshold which improves the previously known results in the wide range of the…