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Two extensions of Ramsey's theorem
Ramsey's theorem, in the version of Erd\H{o}s and Szekeres, states that every
2-coloring of the edges of the complete graph on {1, 2,...,n} contains a
monochromatic clique of order 1/2\log n. In this paper, we consider two
well-studied extensions of Ramsey's theorem.
Improving a result of R\"odl, we show that there is a constant such
that every 2-coloring of the edges of the complete graph on \{2, 3,...,n\}
contains a monochromatic clique S for which the sum of 1/\log i over all
vertices i \in S is at least c\log\log\log n. This is tight up to the constant
factor c and answers a question of Erd\H{o}s from 1981.
Motivated by a problem in model theory, V\"a\"an\"anen asked whether for
every k there is an n such that the following holds. For every permutation \pi
of 1,...,k-1, every 2-coloring of the edges of the complete graph on {1, 2,
..., n} contains a monochromatic clique a_1<...<a_k with
a_{\pi(1)+1}-a_{\pi(1)}>a_{\pi(2)+1}-a_{\pi(2)}>...>a_{\pi(k-1)+1}-a_{\pi(k-1)}.
That is, not only do we want a monochromatic clique, but the differences
between consecutive vertices must satisfy a prescribed order. Alon and,
independently, Erd\H{o}s, Hajnal and Pach answered this question affirmatively.
Alon further conjectured that the true growth rate should be exponential in k.
We make progress towards this conjecture, obtaining an upper bound on n which
is exponential in a power of k. This improves a result of Shelah, who showed
that n is at most double-exponential in k.Comment: 21 pages, accepted for publication in Duke Math.
Open questions about Ramsey-type statements in reverse mathematics
Ramsey's theorem states that for any coloring of the n-element subsets of N
with finitely many colors, there is an infinite set H such that all n-element
subsets of H have the same color. The strength of consequences of Ramsey's
theorem has been extensively studied in reverse mathematics and under various
reducibilities, namely, computable reducibility and uniform reducibility. Our
understanding of the combinatorics of Ramsey's theorem and its consequences has
been greatly improved over the past decades. In this paper, we state some
questions which naturally arose during this study. The inability to answer
those questions reveals some gaps in our understanding of the combinatorics of
Ramsey's theorem.Comment: 15 page
On-line Ramsey numbers
Consider the following game between two players, Builder and Painter. Builder
draws edges one at a time and Painter colours them, in either red or blue, as
each appears. Builder's aim is to force Painter to draw a monochromatic copy of
a fixed graph G. The minimum number of edges which Builder must draw,
regardless of Painter's strategy, in order to guarantee that this happens is
known as the on-line Ramsey number \tilde{r}(G) of G. Our main result, relating
to the conjecture that \tilde{r}(K_t) = o(\binom{r(t)}{2}), is that there
exists a constant c > 1 such that \tilde{r}(K_t) \leq c^{-t} \binom{r(t)}{2}
for infinitely many values of t. We also prove a more specific upper bound for
this number, showing that there exists a constant c such that \tilde{r}(K_t)
\leq t^{-c \frac{\log t}{\log \log t}} 4^t. Finally, we prove a new upper bound
for the on-line Ramsey number of the complete bipartite graph K_{t,t}.Comment: 11 page
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