89 research outputs found
Monochromatic loose paths in multicolored -uniform cliques
For integers and , a -uniform hypergraph is called a
loose path of length , and denoted by , if it consists of
edges such that if and
if . In other words, each pair of
consecutive edges intersects on a single vertex, while all other pairs are
disjoint. Let be the minimum integer such that every
-edge-coloring of the complete -uniform hypergraph yields a
monochromatic copy of . In this paper we are mostly interested in
constructive upper bounds on , meaning that on the cost of
possibly enlarging the order of the complete hypergraph, we would like to
efficiently find a monochromatic copy of in every coloring. In
particular, we show that there is a constant such that for all ,
, , and , there is an
algorithm such that for every -edge-coloring of the edges of , it
finds a monochromatic copy of in time at most . We also
prove a non-constructive upper bound
Rainbow Generalizations of Ramsey Theory - A Dynamic Survey
In this work, we collect Ramsey-type results concerning rainbow edge colorings of graphs
The history of degenerate (bipartite) extremal graph problems
This paper is a survey on Extremal Graph Theory, primarily focusing on the
case when one of the excluded graphs is bipartite. On one hand we give an
introduction to this field and also describe many important results, methods,
problems, and constructions.Comment: 97 pages, 11 figures, many problems. This is the preliminary version
of our survey presented in Erdos 100. In this version 2 only a citation was
complete
Gallai-Ramsey and vertex proper connection numbers
Given a complete graph G, we consider two separate scenarios. First, we consider the minimum number N such that every coloring of G using exactly k colors contains either a rainbow triangle or a monochromatic star on t vertices. This number is known for small cases and generalized for larger cases for a fixed k. Second, we introduce the vertex proper connection number of a graph and provide a relationship to the chromatic number of minimally connected subgraphs. Also a notion of total proper connection is introduced and a question is asked about a possible relationship between the total proper connection number and the vertex and edge proper connection numbers
- …