Related papers: On Monochromatic Solutions of Linear Equations Usi…
The Rado number of an equation is a Ramsey-theoretic quantity associated to the equation. Let $\mathcal{E}$ be a linear equation. Denote by $\operatorname{R}_r(\mathcal{E})$ the minimal integer, if it exists, such that any $r$-coloring of…
In this short note, we present a purely entropic proof that in a $3$-edge-colored simple graph with $R$ red edges, $G$ green edges, and $B$ blue edges, the number of rainbow triangles is at most $\sqrt{2RGB}$.
Kosek, Robertson, Sabo, and Schaal studied the minimum number \(M_k(n)\) of monochromatic solutions to the strict Schur inequality system $x_1\le x_2\le x_3$ and $x_1+x_2<x_3$ in \(2\)-colorings of \([k+1,k+n]\). They proved that for every…
A conjecture of Erd\H{o}s, Graham, Montgomery, Rothschild, Spencer and Straus states that, with the exception of equilateral triangles, any two-coloring of the plane will have a monochromatic congruent copy of every three-point…
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…
In this paper, we study a Ramsey-type problem for equations of the form $ax+by=p(z)$. We show that if certain technical assumptions hold, then any 2-colouring of the positive integers admits infinitely many monochromatic solutions to the…
A system of homogeneous linear equations with integer coefficients is partition regular if, whenever the natural numbers are finitely coloured, the system has a monochromatic solution. The Finite Sums theorem provided the first example of…
Hindman proved that, whenever the set $\mathbb{N}$ of naturals is finitely colored, there must exist non-constant monochromatic solution of the equation $a+b=cd$. In this paper we extend this result for dense subsemigroups of $((0, \infty),…
We prove that there are $O(n)$ tangencies among any set of $n$ red and blue planar curves in which every pair of curves intersects at most once and no two curves of the same color intersect. If every pair of curves may intersect more than…
We show that if we color the hyperedges of the complete $3$-uniform complete graph on $2n+\sqrt{18n+1}+2$ vertices with $n$ colors, then one of the color classes contains a loose path of length three.
We give a short, explicit proof of Hindman's Theorem that in every finite coloring of the integers, there is an infinite set all of whose finite sums have the same color. We give several exampls of colorings of the integers which do not…
A majority coloring of an undirected graph is a vertex coloring in which for each vertex there are at least as many bi-chromatic edges containing that vertex as monochromatic ones. It is known that for every countable graph a majority…
A question posed independently by Letzter and Pokrovskiy asks: how many vertex-disjoint monochromatic cycles are needed to cover the vertex set of an $r$-edge-coloured graph, as a function of its minimum (uncoloured) degree? We resolve this…
The Hadwiger-Nelson problem asks for the minimum number of colors, so that each point of the plane can be assigned a single color with the property that no two points unit-distance apart are identically colored. It is now known that the…
A classic result of Erd\H{o}s, Gy\'arf\'as and Pyber states that for every coloring of the edges of $K_n$ with $r$ colors, there is a cover of its vertex set by at most $f(r) = O(r^2 \log r)$ vertex-disjoint monochromatic cycles. In…
It is consistent for every (1 <= n< omega) that (2^omega = omega_n) and there is a function (F:[omega_n]^{< omega}-> omega) such that every finite set can be written at most (2^n-1) ways as the union of two distinct monocolored sets. If GCH…
List colouring is an NP-complete decision problem even if the total number of colours is three. It is hard even on planar bipartite graphs. We give a polynomial-time algorithm for solving list colouring of permutation graphs with a bounded…
In this work, we consider a number of problems defined on the triangular lattice with $n$ rows, which we will denote as $T_n$. Define a \textit{proper coloring} to be an assignment of colors to the points of $T_n$ such that no three points…
We present a recursive formula for the number of ways to color $j$ vertices blue in an r-uniform hyperpath of size $n$ while avoiding a blue monochromatic sub-hyperpath of length k. We use this result to solve the corresponding problem for…
We present a short ultrafilter proof of the existence of monochromatic exponential triples $\{a, b, b^a\}$ in any finite coloring of the natural numbers. The proof is given from scratch and uses only Ramsey's theorem, the notion of…