Related papers: Ramsey Functions for Generalized Progressions

A geometric progression of length $k$ and integer ratio is a set of numbers of the form $\{a,ar,\dots,ar^{k-1}\}$ for some positive real number $a$ and integer $r\geq 2$. For each integer $k \geq 3$, a greedy algorithm is used to construct…

A strictly increasing sequence of positive integers is called a slightly curved sequence with small error if the sequence can be well-approximated by a function whose second derivative goes to zero faster than or equal to $1/x^\alpha$ for…

One of the toughest problems in Ramsey theory is to determine the existence of monochromatic arithmetic progressions in groups whose elements have been colored. We study the harder problem to not only determine the existence of…

Let $r$ and $k$ be positive integers with $r \mid k$. Denote by $w_{\mathrm{\mathfrak{z}}}(k;r)$ the minimum integer such that every coloring $\chi:[1,w_{\mathrm{\mathfrak{z}}}(k;r)] \rightarrow \{0,1,\dots,r-1\}$ admits a $k$-term…

Van der Waerden's theorem states that for any positive integers $k$ and $r$, there exists a smallest value $n = w(k,r)$, called the van der Waerden number, such that every $r$-coloring of $\{1,\dots,n\}$ contains a monochromatic $k$-term…

In this paper we prove a basic theorem which says that if f : F_p^n -> [0,1] has the property that ||f^||_(1/3) is not too ``large''(actually, it also holds for quasinorms 1/2-\delta in place of 1/3), and E(f) = p^{-n} sum_m f(m) is not too…

The purpose of this survey is to provide a gentle introduction to several recent breakthroughs in graph Ramsey theory. In particular, we will outline the proofs (due to various groups of authors) of exponential improvements to the diagonal,…

The classical hypergraph Ramsey number $r_k(s,n)$ is the minimum $N$ such that for every red-blue coloring of the $k$-tuples of $\{1,\ldots, N\}$, there are $s$ integers such that every $k$-tuple among them is red, or $n$ integers such that…

In this paper, we study $k$-term arithmetic progressions $N, N+d, ..., N+(k-1)d$ of powerful numbers. Under the $abc$-conjecture, we obtain $d \gg_\epsilon N^{1/2 - \epsilon}$. On the other hand, there exist infinitely many $3$-term…

A sequence of graphs $ \{G_k\} $ is a Ramsey sequence if for every positive integer $ k $, the graph $ G_k $ is a proper subgraph of $ G_{k+1} $, and there exists an integer $n > k$ such that every red-blue coloring of $ G_n $ contains a…

Ramsey theory is a central and active branch of combinatorics. Although Ramsey numbers for graphs have been extensively investigated since Ramsey's work in the 1930s, there is still an exponential gap between the best known lower and upper…

We study arithmetic progressions in primes with common differences as small as possible. Tao and Ziegler showed that, for any $k \geq 3$ and $N$ large, there exist non-trivial $k$-term arithmetic progressions in (any positive density subset…

Van der Waerden's classical theorem on arithmetic progressions states that for any positive integers k and r, there exists a least positive integer, w(k,r), such that any r-coloring of {1,2,...,w(k,r)} must contain a monochromatic k-term…

In [5] Graham and Rothschild consider a geometric Ramsey problem: finding the least n such that if all edges of the complete graph on the points {+1,-1}^n are 2-colored, there exist 4 coplanar points such that the 6 edges between them are…

Let H_1, ..., H_k be graphs. The multicolor Ramsey number r(H_1,...,H_k) is the minimum integer r such that in every edge-coloring of K_r by k colors, there is a monochromatic copy of H_i in color i for some 1 <= i <= k. In this paper, we…

We develop novel techniques which allow us to prove a diverse range of results relating to subset sums and complete sequences of positive integers, including solutions to several longstanding open problems. These include: solutions to the…

In this paper, we study the rainbow Erd\H{o}s-Rothschild problem with respect to $k$-term arithmetic progressions. For a set of positive integers $S \subseteq [n]$, an $r$-coloring of $S$ is \emph{rainbow $k$-AP-free} if it contains no…

For an integer $b \geqslant 2$ and a set $S\subset \{0,\cdots,b-1\}$, we define the Kempner set $\mathcal{K}(S,b)$ to be the set of all non-negative integers whose base-$b$ digital expansions contain only digits from $S$. These well-studied…

We prove a known 2-coloring of the integers $[N] := \{1,2,3,\ldots,N\}$ minimizes the number of monochromatic arithmetic 3-progressions under certain restrictions. A monochromatic arithmetic progression is a set of equally-spaced integers…

We investigate the distributions of the number of: (1) monochromatic complete subgraphs over edgewise 2-colorings of complete graphs; and (2) monochromatic arithmetic progressions over 2-colorings of intervals, as statistical Ramsey theory…