Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: An exponential-type upper bound for Folkman numbers   For given integers $k$ and $r$, the Folkman number $f(k;r)$ is the smallest
number of vertices in a graph $G$ which contains no clique on $k+1$ vertices,
yet for every partition of its edges into $r$ parts, some part contains a
clique of order $k$. The existence (finiteness) of Folkman numbers was
established by Folkman (1970) for $r=2$ and by Ne\v{s}et\v{r}il and R\"odl
(1976) for arbitrary $r$, but these proofs led to very weak upper bounds on
$f(k;r)$.
  Recently, Conlon and Gowers and independently the authors obtained a doubly
exponential bound on $f(k;2)$. Here, we establish a further improvement by
showing an upper bound on $f(k;r)$ which is exponential in a polynomial
function of $k$ and $r$. This is comparable to the known lower bound
$2^{\Omega(rk)}$.
  Our proof relies on a recent result of Saxton and Thomason (2015) (or,
alternatively, on a recent result of Balogh, Morris, and Samotij (2015)) from
which we deduce a quantitative version of Ramsey's theorem in random graphs.

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The categories of this paper are: nucl-ex
20
The categories of this paper are: acc-phys
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The categories of this paper are: quant-ph
27
The categories of this paper are: comp-gas
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The categories of this paper are: eess
17
The categories of this paper are: q-fin
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The categories of this paper are: q-alg
7
The categories of this paper are: hep-th
36
The categories of this paper are: cmp-lg
5
The categories of this paper are: gr-qc
35
The categories of this paper are: mtrl-th
33
The categories of this paper are: plasm-ph
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The categories of this paper are: hep-lat
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The categories of this paper are: econ
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The categories of this paper are: chao-dyn
34
The categories of this paper are: chem-ph