Related papers: Almost Empty Monochromatic Triangles in Planar Poi…

We created and parallelized two SAT solvers to find new bounds on some Ramsey-type numbers. For $c > 0$, let $R_c(L)$ be the least $n$ such that for all $c$-colorings of the $[n]\times [n]$ lattice grid there will exist a monochromatic…

We discuss a problem posed by Ronald Graham about the minimum number, over all 2-colorings of $[1,n]$, of monochromatic $\{x,y,x+ay\}$ triples for $a \geq 1$. We give a new proof of the original case of $a=1$. We show that the minimum…

Let k, r, s in the natural numbers where r \geq s \geq 2. Define f(s,r,k) to be the smallest positive integer n such that for every coloring of the integers in [1,n] there exist subsets S_1 and S_2 such that: (a) S_1 and S_2 are…

Montassier, Raspaud, and Wang (2006) asked to find the smallest positive integers $d_0$ and $d_1$ such that planar graphs without $\{4,5\}$-cycles and $d^{\Delta}\ge d_0$ are $3$-choosable and planar graphs without $\{4,5,6\}$-cycles and…

A graph is (m,k)-colourable if its vertices can be coloured with m colours such that the maximum degree of the subgraph induced on the set of all vertices receiving the same colour is at most k. The k-defective chromatic number $\chi_k(G)$…

We prove that, for positive integers $n,a_1, a_2, a_3$ satisfying $a_1+a_2+a_3 = n-1$, it holds that any bipartite graph $G$ which is the union of three perfect matchings $M_1$, $M_2$, and $M_3$ on $2n$ vertices contains a matching $M$ such…

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…

For all non-degenerate triangles T, we determine the minimum number of colors needed to color the plane such that no max-norm isometric copy of T is monochromatic.

Let $S$ be a set of $n$ points in general position in the plane. The Second Selection Lemma states that for any family of $\Theta(n^3)$ triangles spanned by $S$, there exists a point of the plane that lies in a constant fraction of them.…

We say that $G$ is a $(3, 3)$-Ramsey graph if every $2$-coloring of the edges of $G$ forces a monochromatic triangle. The $(3, 3)$-Ramsey graph $G$ is minimal if $G$ does not contain a proper $(3, 3)$-Ramsey subgraph. In this work we find…

A graph is (m, k)-colourable if its vertices can be coloured with m colours such that the maximum degree of any subgraph induced on ver- tices receiving the same colour is at most k. The k-defective chromatic number for a graph is the least…

Let $[a,b]$ denote the integers between $a$ and $b$ inclusive and, for a finite subset $X \subseteq \mathbb{Z}$, let the diameter of $X$ be equal to $\max(X)-\min(X)$. We write $X<_p\,Y$ provided $\max(X)<\min(Y)$. For a positive integer…

A weak $c$-colouring of a balanced incomplete block design (BIBD) is a colouring of the points of the design with $c$ colours in such a way that no block of the design has all of its vertices receive the same colour. A BIBD is said to be…

For integers $n\ge 0$, an iterated triangulation $Tr(n)$ is defined recursively as follows: $Tr(0)$ is the plane triangulation on three vertices and, for $n\ge 1$, $Tr(n)$ is the plane triangulation obtained from the plane triangulation…

For positive integers $n,d$, let $\lambda(n,d)$ be the minimal number of vertices of a triangulation of $n$-sphere which admits a degree $d$ simplicial map to the boundary of $(n+1)$-simplex. We show that…

For planar graphs, we consider the problems of \emph{list edge coloring} and \emph{list total coloring}. Edge coloring is the problem of coloring the edges while ensuring that two edges that are adjacent receive different colors. Total…

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…

In the Minimum Consistent Subset (MCS) problem, we are presented with a connected simple undirected graph $G=(V,E)$, consisting of a vertex set $V$ of size $n$ and an edge set $E$. Each vertex in $V$ is assigned a color from the set…

For all positive integers $r\geq 3$ and $n$ such that $r^2-r$ divides $n$ and an affine plane of order $r$ exists, we construct an $r$-edge colored graph with minimum degree $(1-\frac{r-2}{r^2-r})n-2$ such that the largest monochromatic…

Let $S$ be an orthogonal array $OA(d,k)$ and let $c$ be an $r$--coloring of its ground set $X$. We give a combinatorial identity which relates the number of vectors in $S$ with given color patterns under $c$ with the cardinalities of the…