Related papers: A sparse canonical van der Waerden theorem

We present a self-contained proof of a strong version of van der Waerden's Theorem. By using translation invariant filters that are maximal with respect to inclusion, a simple inductive argument shows the existence of "piecewise…

Given a graph $H$, let $g(n,H)$ denote the smallest $k$ for which the following holds. We can assign a $k$-colouring $f_v$ of the edge set of $K_n$ to each vertex $v$ in $K_n$ with the property that for any copy $T$ of $H$ in $K_n$, there…

A celebrated result of R\"odl and Ruci\'nski states that for every graph $F$, which is not a forest of stars and paths of length $3$, and fixed number of colours $r\ge 2$ there exist positive constants $c, C$ such that for $p \leq…

In this work, we develop a unified framework for establishing sharp threshold results for various Ramsey properties. To achieve this, we view such properties as non-colourability of auxiliary hypergraphs. Our main technical result gives…

We study an anti-Ramsey extension of the classical Corr\'{a}di--Hajnal Theorem: how many colors are needed to color the complete graph on $n$ vertices in order to guarantee a rainbow copy of $t K_{3}$, that is, $t$ vertex-disjoint…

In this article, we investigate polynomial generalizations of the van der Waerden theorem with a focus on largeness properties of recurrence patterns. We prove an $IP_r^\star$-strengthened version of the polynomial van der Waerden theorem,…

Van der Waerden's theorem asserts that if you color the natural numbers with, say, five different colors, then you can always find arbitrarily long sequences of numbers that have the same color and that form an arithmetic progression.…

Answering a question raised by Dudek and Pra\l{}at, we show that if $pn\rightarrow \infty$, w.h.p.,~whenever $G=G(n,p)$ is $2$-coloured, there exists a monochromatic path of length $n(2/3+o(1))$. This result is optimal in the sense that…

We present a new short proof of Van der Waerden's Theorem about the existence of arbitrarily long monochromatic arithmetic progressions. The proof uses algebra in the compact space of ultrafilters $\beta\N$, but contrarily to the other…

For any countably infinite graph $G$, Ramsey's theorem guarantees an infinite monochromatic copy of $G$ in any $r$-coloring of the edges of the countably infinite complete graph $K_\mathbb{N}$. Taking this a step further, it is natural to…

Extending Furstenberg's ergodic theoretic proof for Szemer\'edi's theorem on arithmetic progressions, Furstenberg and Weiss (2003) proved the following qualitative result. For every d and k, there exists an integer N such that no matter how…

We study rainbow-free colourings of $k$-uniform hypergraphs; that is, colourings that use $k$ colours but with the property that no hyperedge attains all colours. We show that $p^*=(k-1)(\ln n)/n$ is the threshold function for the existence…

We show that in every two-colouring of the edges of the complete graph $K_N$ there is a monochromatic $K_k$ which can be extended in at least $(1 + o_k(1))2^{-k}N$ ways to a monochromatic $K_{k+1}$. This result is asymptotically best…

A sequence of positive integers $w_1,w_2,...,w_n$ is called an ascending wave if $w_{i+1}-w_i \geq w_i - w_{i-1}$ for $2 \leq i \leq n-1$. For integers $k,r\geq1$, let $AW(k;r)$ be the least positive integer such that under any $r$-coloring…

We show how to adjust a very nice coupling argument due to McDiarmid in order to prove/reprove in a novel way results concerning Hamilton cycles in various models of random graph and hypergraphs. In particular, we firstly show that for…

Given a graph $G$, an exact $r$-coloring of $G$ is a surjective function $c:V(G) \to [1,\dots,r]$. An arithmetic progression in $G$ of length $j$ with common difference $d$ is a set of vertices $\{v_1,\dots, v_j\}$ such that…

A $\textit{ladder}$ is a set $S \subseteq \mathbb Z^+$ such that any finite coloring of $\mathbb Z$ contains arbitrarily long monochromatic progressions with common difference in $S$. Van der Waerden's theorem famously asserts that $\mathbb…

We investigate Ramsey properties of a random graph model in which random edges are added to a given dense graph. Specifically, we determine lower and upper bounds on the function $p=p(n)$ that ensures that for any dense graph $G_n$ a.a.s.…

Let $G_{n,p}^{[\kappa]}$ denote the space of $n$-vertex edge coloured graphs, where each edge occurs independently with probability $p$. The colour of each existing edge is chosen independently and uniformly at random from the set…

A classical result of Erd\H{o}s and Hajnal claims that for any integers $k, r, g \geq 2$ there is an $r$-uniform hypergraph of girth at least $g$ with chromatic number at least $k$. This implies that there are sparse hypergraphs such that…