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The Ramsey size number of dipaths
AbstractLet H be a finite graph. The Ramsey size number of H, r̂(H,H), is the minimum number of edges required to construct a graph such that when its edges are 2-colored it contains a monochromatic subgraph, H. In this paper we prove that the Ramsey size number of a directed path is quadratic
Directed Ramsey number for trees
In this paper, we study Ramsey-type problems for directed graphs. We first
consider the -colour oriented Ramsey number of , denoted by
, which is the least for which every
-edge-coloured tournament on vertices contains a monochromatic copy of
. We prove that for any oriented
tree . This is a generalisation of a similar result for directed paths by
Chv\'atal and by Gy\'arf\'as and Lehel, and answers a question of Yuster. In
general, it is tight up to a constant factor.
We also consider the -colour directed Ramsey number
of , which is defined as above, but, instead
of colouring tournaments, we colour the complete directed graph of order .
Here we show that for any
oriented tree , which is again tight up to a constant factor, and it
generalises a result by Williamson and by Gy\'arf\'as and Lehel who determined
the -colour directed Ramsey number of directed paths.Comment: 27 pages, 14 figure
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
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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