Related papers: Large monochromatic components in 3-edge-colored S…
A weighting of the edges of a hypergraph is called vertex-coloring if the weighted degrees of the vertices yield a proper coloring of the graph, i.e., every edge contains at least two vertices with different weighted degrees. In this paper…
Let $\R$ be the set of all finite graphs $G$ with the Ramsey property that every coloring of the edges of $G$ by two colors yields a monochromatic triangle. In this paper we establish a sharp threshold for random graphs with this property.…
A colouring of a hypergraph's vertices is polychromatic if every hyperedge contains at least one vertex of each colour; the polychromatic number is the maximum number of colours in such a colouring. Its dual, the cover-decomposition number,…
A well-studied coloring problem is to assign colors to the edges of a graph $G$ so that, for every pair of vertices, all edges of at least one shortest path between them receive different colors. The minimum number of colors necessary in…
This paper begins by exploring some old and new results about Ramsey numbers and minimum numbers of monochromatic triangles in $2$-colorings of complete graphs, both in the disjoint and non-disjoint cases. We then extend the theory, by…
We prove that the statement "for every infinite cardinal nu, every graph with list chromatic nu has coloring number at most beth_omega (nu)" proved by Kojman [6] using the RGCH theorem [11] implies the RGCG theorem via a short forcing…
It was previously shown that any two-colour colouring of K(C_n) must contain a monochromatic planar K_4 subgraph for n >= N^*, where 6 <= N^* <= N and N is Graham's number. The bound was later improved to 11 <= N^* <= N. In this article, it…
We show an $\Omega\big(\Delta^{\frac{1}{3}-\frac{\eta}{3}}\big)$ lower bound on the runtime of any deterministic distributed $\mathcal{O}\big(\Delta^{1+\eta}\big)$-graph coloring algorithm in a weak variant of the \LOCAL\ model. In…
Given a family of 3-graphs F its codegree threshold coex(n, F) is the largest number d=d(n) such that there exists an n-vertex 3-graph in which every pair of vertices is contained in at least d 3-edges but which contains no member of F as a…
We study a Tur\'an-type problem on edge-colored complete graphs. We show that for any $r$ and $t$, any sufficiently large $r$-edge-colored complete graph on $n$ vertices with $\Omega(n^{2-1/tr^r})$ edges in each color contains a member from…
An island in a graph is a set $X$ of vertices, such that each element of $X$ has few neighbors outside $X$. In this paper, we prove several bounds on the size of islands in large graphs embeddable on fixed surfaces. As direct consequences…
A proper total colouring of a graph $G$ is called harmonious if it has the further property that when replacing each unordered pair of incident vertices and edges with their colours, then no pair of colours appears twice. The smallest…
Since planar triangle-free graphs are 3-colourable, such a graph with n vertices has an independent set of size at least n/3. We prove that unless the graph contains a certain obstruction, its independence number is at least n/(3-epsilon)…
We settle a problem of Havel by showing that there exists an absolute constant d such that if G is a planar graph in which every two distinct triangles are at distance at least d, then G is 3-colorable. In fact, we prove a more general…
Let F be a family of r-uniform hypergraphs. The chromatic threshold of F is the infimum of all non-negative reals c such that the subfamily of F comprising hypergraphs H with minimum degree at least $c \binom{|V(H)|}{r-1}$ has bounded…
We say that a signed graph is $k$-critical if it is not $k$-colorable but every one of its proper subgraphs is $k$-colorable. Using the definition of colorability due to Naserasr, Wang, and Zhu that extends the notion of circular…
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…
We show that the threshold for the binomial random $3$-partite, $3$-uniform hypergraph $G^{3}((n,n,n),p)$ to contain a Latin square is $\Theta(\log{n}/n)$. We also prove analogous results for Steiner triple systems and proper list…
A tree $T$ in an edge-colored graph is a {\it proper tree} if no two adjacent edges of $T$ receive the same color. Let $G$ be a connected graph of order $n$ and $k$ be a fixed integer with $2\le k\le n$. For a vertex subset $S \subseteq…
In this note, we improve on results of Hoppen, Kohayakawa and Lefmann about the maximum number of edge colorings without monochromatic copies of a star of a fixed size that a graph on $n$ vertices may admit. Our results rely on an improved…