886 research outputs found
Monotone paths in random hypergraphs
We determine the probability thresholds for the existence of monotone paths,
of finite and infinite length, in random oriented graphs with vertex set
, the set of all increasing -tuples in . These
graphs appear as line graph of uniform hypergraphs with vertex set .Comment: 16 page
Ramsey numbers of ordered graphs
An ordered graph is a pair where is a graph and
is a total ordering of its vertices. The ordered Ramsey number
is the minimum number such that every ordered
complete graph with vertices and with edges colored by two colors contains
a monochromatic copy of .
In contrast with the case of unordered graphs, we show that there are
arbitrarily large ordered matchings on vertices for which
is superpolynomial in . This implies that
ordered Ramsey numbers of the same graph can grow superpolynomially in the size
of the graph in one ordering and remain linear in another ordering.
We also prove that the ordered Ramsey number is
polynomial in the number of vertices of if the bandwidth of
is constant or if is an ordered graph of constant
degeneracy and constant interval chromatic number. The first result gives a
positive answer to a question of Conlon, Fox, Lee, and Sudakov.
For a few special classes of ordered paths, stars or matchings, we give
asymptotically tight bounds on their ordered Ramsey numbers. For so-called
monotone cycles we compute their ordered Ramsey numbers exactly. This result
implies exact formulas for geometric Ramsey numbers of cycles introduced by
K\'arolyi, Pach, T\'oth, and Valtr.Comment: 29 pages, 13 figures, to appear in Electronic Journal of
Combinatoric
Combinatorial theorems relative to a random set
We describe recent advances in the study of random analogues of combinatorial
theorems.Comment: 26 pages. Submitted to Proceedings of the ICM 201
Erdos-Szekeres-type theorems for monotone paths and convex bodies
For any sequence of positive integers j_1 < j_2 < ... < j_n, the k-tuples
(j_i,j_{i + 1},...,j_{i + k-1}), i=1, 2,..., n - k+1, are said to form a
monotone path of length n. Given any integers n\ge k\ge 2 and q\ge 2, what is
the smallest integer N with the property that no matter how we color all
k-element subsets of [N]=\{1,2,..., N\} with q colors, we can always find a
monochromatic monotone path of length n? Denoting this minimum by N_k(q,n), it
follows from the seminal 1935 paper of Erd\H os and Szekeres that
N_2(q,n)=(n-1)^q+1 and N_3(2,n) = {2n -4\choose n-2} + 1. Determining the other
values of these functions appears to be a difficult task. Here we show that
2^{(n/q)^{q-1}} \leq N_3(q,n) \leq 2^{n^{q-1}\log n}, for q \geq 2 and n \geq
q+2. Using a stepping-up approach that goes back to Erdos and Hajnal, we prove
analogous bounds on N_k(q,n) for larger values of k, which are towers of height
k-1 in n^{q-1}. As a geometric application, we prove the following extension of
the Happy Ending Theorem. Every family of at least M(n)=2^{n^2 \log n} plane
convex bodies in general position, any pair of which share at most two boundary
points, has n members in convex position, that is, it has n members such that
each of them contributes a point to the boundary of the convex hull of their
union.Comment: 32 page
Metric Construction, Stopping Times and Path Coupling
In this paper we examine the importance of the choice of metric in path
coupling, and the relationship of this to \emph{stopping time analysis}. We
give strong evidence that stopping time analysis is no more powerful than
standard path coupling. In particular, we prove a stronger theorem for path
coupling with stopping times, using a metric which allows us to restrict
analysis to standard one-step path coupling. This approach provides insight for
the design of non-standard metrics giving improvements in the analysis of
specific problems.
We give illustrative applications to hypergraph independent sets and SAT
instances, hypergraph colourings and colourings of bipartite graphs.Comment: 21 pages, revised version includes statement and proof of general
stopping times theorem (section 2.2), and additonal remarks in section
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