Related papers: A Generalization of Schur's Theorem

Answering a conjecture of A. Sisto, J. Sahasrabudhe proved the exponential version of the Schur theorem: for every finite coloring of the naturals, there exists a monochromatic copy of $\{x,y,x^y:x\neq y\},$ which initiates the study of…

For two graphs, $G$ and $F$, and an integer $r\ge2$ we write $G\rightarrow (F)_r$ if every $r$-coloring of the edges of $G$ results in a monochromatic copy of $F$. In 1995, the first two authors established a threshold edge probability for…

A standard proof of Schur's Theorem yields that any $r$-coloring of $\{1,2,\dots,R_r-1\}$ yields a monochromatic solution to $x+y=z$, where $R_r$ is the classical $r$-color Ramsey number, the minimum $N$ such that any $r$-coloring of a…

A well-known result of Burr, Erd\H{o}s and Spencer [Transactions of the American Mathematical Society, 1975] determines the $2$-colour Ramsey number for any sufficiently large collection of vertex-disjoint copies of a fixed graph $H$…

We start with a bijective proof of Schur's theorem due to Alladi and Gordon and describe how a particular iteration of it leads to some very general theorems on colored partitions. These theorems imply a number of important results,…

The Ramsey number $r(G)$ of a graph $G$ is the minimum $N$ such that every red-blue coloring of the edges of the complete graph on $N$ vertices contains a monochromatic copy of $G$. Determining or estimating these numbers is one of the…

W. T. Gower generalized Hindman's Finite sum theorem over $X_{k}=\left\{ \left(n_{1},n_{2},\ldots,n_{k}\right):n_{1}\neq0\right\} $ by showing that for any finite coloring of $X_{k}$ there exists a sequence such that the Gower subspace…

Superfilters are generalized ultrafilters, which capture the underlying concept in Ramsey theoretic theorems such as van der Waerden's Theorem. We establish several properties of superfilters, which generalize both Ramsey's Theorem and its…

R\"odl and Ruci\'nski (1990) established Ramsey's theorem for random graphs. In particular, for fixed integers $r$, $\ell\geq 2$ they showed that $\hat p_{K_\ell,r}(n)=n^{-\frac{2}{\ell+1}}$ is a threshold for the Ramsey property that every…

In 1967, Gerencs\'er and Gy\'arf\'as proved a result which is considered the starting point of graph-Ramsey theory: In every 2-coloring of $K_n$ there is a monochromatic path on $\lceil(2n+1)/3\rceil$ vertices, and this is best possible.…

We resolve the Ramsey problem for $\{x,y,z:x+y=p(z)\}$ for all polynomials $p$ over $\mathbb{Z}$. In particular, we characterise all polynomials that are $2$-Ramsey, that is, those $p(z)$ such that any $2$-colouring of $\mathbb{N}$ contains…

We show that, for every $r, k$, there is an $n = n(r,k)$ so that any $r$-coloring of the edges of the complete graph on $[n]$ will yield a monochromatic complete subgraph on vertices ${a + \sum_{i \in I} d_i \mid I \subseteq [k]}$ for some…

In this note we study graphs $G_r$ with the property that every colouring of $E(G_r)$ with $r+1$ colours admits a copy of some graph $H$ using at most $r$ colours. For $1\le r\le e(H)$ such graphs occur naturally at intermediate steps in…

We give a short proof of the following theorem due to Jon H. Folkman (1969): The chromatic number of any graph is at most $2$ plus the maximum over all subgraphs of the difference between half the number of vertices and the independence…

We consider $m$-colorings of the edges of a complete graph, where each color class is defined semi-algebraically with bounded complexity. The case $m = 2$ was first studied by Alon et al., who applied this framework to obtain surprisingly…

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…

Hindman's celebrated Finite Sums Theorem, and its high-dimensional version due to Milliken and Taylor, are extended from covers of countable sets to covers of arbitrary topological spaces with Menger's classic covering property. The methods…

We prove the Schr\"oder case, i.e. the case $\langle \cdot,e_{n-d}h_d \rangle$, of the conjecture of Haglund, Remmel and Wilson (Haglund et al. 2018) for $\Delta_{h_m}\Delta_{e_{n-k-1}}'e_n$ in terms of decorated partially labelled Dyck…

The Kneser conjecture (1955) was proved by Lov\'asz (1978) using the Borsuk-Ulam theorem; all subsequent proofs, extensions and generalizations also relied on Algebraic Topology results, namely the Borsuk-Ulam theorem and its extensions.…

In this paper we prove a new result about partitioning coloured complete graphs and use it to determine certain Ramsey numbers exactly. The partitioning theorem we prove is that for k at least 1, in every edge colouring of a complete graph…