174 research outputs found
Countable connected-homogeneous digraphs
A digraph is connected-homogeneous if every isomorphism between two finite
connected induced subdigraphs extends to an automorphism of the whole digraph.
In this paper, we completely classify the countable connected-homogeneous
digraphs.Comment: 49 page
Colouring the Triangles Determined by a Point Set
Let P be a set of n points in general position in the plane. We study the
chromatic number of the intersection graph of the open triangles determined by
P. It is known that this chromatic number is at least n^3/27+O(n^2), and if P
is in convex position, the answer is n^3/24+O(n^2). We prove that for arbitrary
P, the chromatic number is at most n^3/19.259+O(n^2)
Hypergraph Ramsey numbers
The Ramsey number r_k(s,n) is the minimum N such that every red-blue coloring
of the k-tuples of an N-element set contains either a red set of size s or a
blue set of size n, where a set is called red (blue) if all k-tuples from this
set are red (blue). In this paper we obtain new estimates for several basic
hypergraph Ramsey problems. We give a new upper bound for r_k(s,n) for k \geq 3
and s fixed. In particular, we show that r_3(s,n) \leq 2^{n^{s-2}\log n}, which
improves by a factor of n^{s-2}/ polylog n the exponent of the previous upper
bound of Erdos and Rado from 1952. We also obtain a new lower bound for these
numbers, showing that there are constants c_1,c_2>0 such that r_3(s,n) \geq
2^{c_1 sn \log (n/s)} for all 4 \leq s \leq c_2n. When s is a constant, it
gives the first superexponential lower bound for r_3(s,n), answering an open
question posed by Erdos and Hajnal in 1972. Next, we consider the 3-color
Ramsey number r_3(n,n,n), which is the minimum N such that every 3-coloring of
the triples of an N-element set contains a monochromatic set of size n.
Improving another old result of Erdos and Hajnal, we show that r_3(n,n,n) \geq
2^{n^{c \log n}}. Finally, we make some progress on related hypergraph
Ramsey-type problems
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