Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Polynomial to exponential transition in Ramsey theory   Given $s \ge k\ge 3$, let $h^{(k)}(s)$ be the minimum $t$ such that there
exist arbitrarily large $k$-uniform hypergraphs $H$ whose independence number
is at most polylogarithmic in the number of vertices and in which every $s$
vertices span at most $t$ edges. Erd\H os and Hajnal conjectured (1972) that
$h^{(k)}(s)$ can be calculated precisely using a recursive formula and Erd\H os
offered \$500 for a proof of this. For $k=3$ this has been settled for many
values of $s$ including powers of three but it was not known for any $k\geq 4$
and $s\geq k+2$.
  Here we settle the conjecture for all $s \ge k \ge 4$. We also answer a
question of Bhat and R\"odl by constructing, for each $k \ge 4$, a quasirandom
sequence of $k$-uniform hypergraphs with positive density and upper density at
most $k!/(k^k-k)$. This result is sharp.

28
The categories of this paper are: alg-geom
18
The categories of this paper are: econ
30
The categories of this paper are: q-alg
4
The categories of this paper are: cond-mat
15
The categories of this paper are: nucl-ex
32
The categories of this paper are: atom-ph
22
The categories of this paper are: chao-dyn
7
The categories of this paper are: hep-th
36
The categories of this paper are: cmp-lg
24
The categories of this paper are: dg-ga
31
The categories of this paper are: ao-sci
19
The categories of this paper are: eess
17
The categories of this paper are: q-fin
21
The categories of this paper are: adap-org
37
The categories of this paper are: supr-con
23
The categories of this paper are: solv-int