11 research outputs found
Coloring triple systems with local conditions
We produce an edge-coloring of the complete 3-uniform hypergraph on n
vertices with colors such that the edges spanned by
every set of five vertices receive at least three distinct colors. This answers
the first open case of a question of Conlon-Fox-Lee-Sudakov [1] who asked
whether such a coloring exists with colors
On locally rainbow colourings
Given a graph , let denote the smallest for which the
following holds. We can assign a -colouring of the edge set of
to each vertex in with the property that for any copy of in
, there is some such that every edge in has a different
colour in .
The study of this function was initiated by Alon and Ben-Eliezer. They
characterized the family of graphs for which is bounded and asked
whether it is true that for every other graph is polynomial. We show
that this is not the case and characterize the family of connected graphs
for which grows polynomially. Answering another question of theirs, we
also prove that for every , there is some
such that for all sufficiently large .
Finally, we show that the above problem is connected to the
Erd\H{o}s-Gy\'arf\'as function in Ramsey Theory, and prove a family of special
cases of a conjecture of Conlon, Fox, Lee and Sudakov by showing that for each
fixed the complete -uniform hypergraph can be edge-coloured
using a subpolynomial number of colours in such a way that at least colours
appear among any vertices.Comment: 12 page
Hypergraph Ramsey numbers of cliques versus stars
Let denote the complete -uniform hypergraph on vertices
and the -uniform hypergraph on vertices consisting of all
edges incident to a given vertex. Whereas many hypergraph Ramsey
numbers grow either at most polynomially or at least exponentially, we show
that the off-diagonal Ramsey number exhibits an
unusual intermediate growth rate, namely, for some positive
constants and . The proof of these bounds brings in a novel Ramsey
problem on grid graphs which may be of independent interest: what is the
minimum such that any -edge-coloring of the Cartesian product contains either a red rectangle or a blue ?Comment: 13 page
Recent developments in graph Ramsey theory
Given a graph H, the Ramsey number r(H) is the smallest natural number N such that any two-colouring of the edges of K_N contains a monochromatic copy of H. The existence of these numbers has been known since 1930 but their quantitative behaviour is still not well understood. Even so, there has been a great deal of recent progress on the study of Ramsey numbers and their variants, spurred on by the many advances across extremal combinatorics. In this survey, we will describe some of this progress
The Erdős-Gyárfás problem on generalized Ramsey numbers
Fix positive integers p and q with [equation; see abstract in PDF for details]. An edge coloring of the complete graph K_n is said to be a (p,q)-coloring if every K_p receives at least q different colors. The function f(n,p,q) is the minimum number of colors that are needed for K_n to have a (p,q)-coloring. This function was introduced about 40 years ago, but Erdős and Gyárfás were the first to study the function in a systematic way. They proved that f(n,p,p) is polynomial in n and asked to determine the maximum q, depending on p, for which f(n,p,q) is subpolynomial in n. We prove that the answer is p - 1
The Erdős-Gyárfás problem on generalized Ramsey numbers
Fix positive integers p and q with [equation; see abstract in PDF for details]. An edge coloring of the complete graph K_n is said to be a (p,q)-coloring if every K_p receives at least q different colors. The function f(n,p,q) is the minimum number of colors that are needed for K_n to have a (p,q)-coloring. This function was introduced about 40 years ago, but Erdős and Gyárfás were the first to study the function in a systematic way. They proved that f(n,p,p) is polynomial in n and asked to determine the maximum q, depending on p, for which f(n,p,q) is subpolynomial in n. We prove that the answer is p - 1