2,049 research outputs found
Short proofs of some extremal results
We prove several results from different areas of extremal combinatorics,
giving complete or partial solutions to a number of open problems. These
results, coming from areas such as extremal graph theory, Ramsey theory and
additive combinatorics, have been collected together because in each case the
relevant proofs are quite short.Comment: 19 page
On the path-avoidance vertex-coloring game
For any graph and any integer , the \emph{online vertex-Ramsey
density of and }, denoted , is a parameter defined via a
deterministic two-player Ramsey-type game (Painter vs.\ Builder). This
parameter was introduced in a recent paper \cite{mrs11}, where it was shown
that the online vertex-Ramsey density determines the threshold of a similar
probabilistic one-player game (Painter vs.\ the binomial random graph
). For a large class of graphs , including cliques, cycles,
complete bipartite graphs, hypercubes, wheels, and stars of arbitrary size, a
simple greedy strategy is optimal for Painter and closed formulas for
are known. In this work we show that for the case where
is a (long) path, the picture is very different. It is not hard to see that
for an appropriately defined integer
, and that the greedy strategy gives a lower bound of
. We construct and analyze Painter strategies that
improve on this greedy lower bound by a factor polynomial in , and we
show that no superpolynomial improvement is possible
Online size Ramsey numbers: Odd cycles vs connected graphs
Given two graph families and , a size Ramsey
game is played on the edge set of . In every round, Builder
selects an edge and Painter colours it red or blue. Builder is trying to force
Painter to create as soon as possible a red copy of a graph from
or a blue copy of a graph from . The online (size) Ramsey number
is the smallest number of rounds in the
game provided Builder and Painter play optimally. We prove that if is the family of all odd cycles and is the family of all
connected graphs on vertices and edges, then , where is the golden
ratio, and for , we have . We also show that
for . As a consequence we get for every .Comment: 14 pages, 0 figures; added appendix containing intuition behind the
potential function used for lower bound; corrected typos and added a few
clarification
Ramsey games with giants
The classical result in the theory of random graphs, proved by Erdos and
Renyi in 1960, concerns the threshold for the appearance of the giant component
in the random graph process. We consider a variant of this problem, with a
Ramsey flavor. Now, each random edge that arrives in the sequence of rounds
must be colored with one of R colors. The goal can be either to create a giant
component in every color class, or alternatively, to avoid it in every color.
One can analyze the offline or online setting for this problem. In this paper,
we consider all these variants and provide nontrivial upper and lower bounds;
in certain cases (like online avoidance) the obtained bounds are asymptotically
tight.Comment: 29 pages; minor revision
Bipartite, Size, and Online Ramsey Numbers of Some Cycles and Paths
The basic premise of Ramsey Theory states that in a sufficiently large system, complete disorder is impossible. One instance from the world of graph theory says that given two fixed graphs F and H, there exists a finitely large graph G such that any red/blue edge coloring of the edges of G will produce a red copy of F or a blue copy of H. Much research has been conducted in recent decades on quantifying exactly how large G must be if we consider different classes of graphs for F and H. In this thesis, we explore several Ramsey- type problems with a particular focus on paths and cycles. We first examine the bipartite size Ramsey number of a path on n vertices, bˆr(Pn), and give an upper bound using a random graph construction motivated by prior upper bound improvements in similar problems. Next, we consider the size Ramsey number Rˆ (C, Pn) and provide a significant improvement to the upper bound using a very structured graph, the cube of a path, as opposed to a random construction. We also prove a small improvement to the lower bound and show that the r-colored version of this problem is asymptotically linear in rn. Lastly, we give an upper bound for the online Ramsey number R˜ (C, Pn)
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
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