24,319 research outputs found
Constrained Ramsey Numbers
For two graphs S and T, the constrained Ramsey number f(S, T) is the minimum
n such that every edge coloring of the complete graph on n vertices, with any
number of colors, has a monochromatic subgraph isomorphic to S or a rainbow
(all edges differently colored) subgraph isomorphic to T. The Erdos-Rado
Canonical Ramsey Theorem implies that f(S, T) exists if and only if S is a star
or T is acyclic, and much work has been done to determine the rate of growth of
f(S, T) for various types of parameters. When S and T are both trees having s
and t edges respectively, Jamison, Jiang, and Ling showed that f(S, T) <=
O(st^2) and conjectured that it is always at most O(st). They also mentioned
that one of the most interesting open special cases is when T is a path. In
this work, we study this case and show that f(S, P_t) = O(st log t), which
differs only by a logarithmic factor from the conjecture. This substantially
improves the previous bounds for most values of s and t.Comment: 12 pages; minor revision
Monochromatic cycle partitions in local edge colourings
An edge colouring of a graph is said to be an -local colouring if the
edges incident to any vertex are coloured with at most colours.
Generalising a result of Bessy and Thomass\'e, we prove that the vertex set of
any -locally coloured complete graph may be partitioned into two disjoint
monochromatic cycles of different colours. Moreover, for any natural number
, we show that the vertex set of any -locally coloured complete graph may
be partitioned into disjoint monochromatic cycles. This
generalises a result of Erd\H{o}s, Gy\'arf\'as and Pyber.Comment: 10 page
Positional Games
Positional games are a branch of combinatorics, researching a variety of
two-player games, ranging from popular recreational games such as Tic-Tac-Toe
and Hex, to purely abstract games played on graphs and hypergraphs. It is
closely connected to many other combinatorial disciplines such as Ramsey
theory, extremal graph and set theory, probabilistic combinatorics, and to
computer science. We survey the basic notions of the field, its approaches and
tools, as well as numerous recent advances, standing open problems and
promising research directions.Comment: Submitted to Proceedings of the ICM 201
The Ramsey Theory of Henson graphs
Analogues of Ramsey's Theorem for infinite structures such as the rationals
or the Rado graph have been known for some time. In this context, one looks for
optimal bounds, called degrees, for the number of colors in an isomorphic
substructure rather than one color, as that is often impossible. Such theorems
for Henson graphs however remained elusive, due to lack of techniques for
handling forbidden cliques. Building on the author's recent result for the
triangle-free Henson graph, we prove that for each , the
-clique-free Henson graph has finite big Ramsey degrees, the appropriate
analogue of Ramsey's Theorem.
We develop a method for coding copies of Henson graphs into a new class of
trees, called strong coding trees, and prove Ramsey theorems for these trees
which are applied to deduce finite big Ramsey degrees. The approach here
provides a general methodology opening further study of big Ramsey degrees for
ultrahomogeneous structures. The results have bearing on topological dynamics
via work of Kechris, Pestov, and Todorcevic and of Zucker.Comment: 75 pages. Substantial revisions in the presentation. Submitte
Two-Source Dispersers for Polylogarithmic Entropy and Improved Ramsey Graphs
In his 1947 paper that inaugurated the probabilistic method, Erd\H{o}s proved
the existence of -Ramsey graphs on vertices. Matching Erd\H{o}s'
result with a constructive proof is a central problem in combinatorics, that
has gained a significant attention in the literature. The state of the art
result was obtained in the celebrated paper by Barak, Rao, Shaltiel and
Wigderson [Ann. Math'12], who constructed a
-Ramsey graph, for some small universal
constant .
In this work, we significantly improve the result of Barak~\etal and
construct -Ramsey graphs, for some universal constant .
In the language of theoretical computer science, our work resolves the problem
of explicitly constructing two-source dispersers for polylogarithmic entropy
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