The Ramsey number
r
k
(
s
,
n
)
r_k(s,n)
is the minimum
N
N
such that every red-blue coloring of the
k
k
-tuples of an
N
N
-element set contains a red set of size
s
s
or a blue set of size
n
n
, where a set is called red (blue) if all
k
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
)
r_k(s,n)
for
k
≥
3
k \geq 3
and
s
s
fixed. In particular, we show that
\[
r
3
(
s
,
n
)
≤
2
n
s
−
2
log
n
,
r_3(s,n) \leq 2^{n^{s-2}\log n},
\]
which improves by a factor of
n
s
−
2
/
polylog
n
n^{s-2}/\textrm {polylog}\,n
the exponent of the previous upper bound of Erdős and Rado from 1952. We also obtain a new lower bound for these numbers, showing that there is a constant
c
>
0
c>0
such that
\[
r
3
(
s
,
n
)
≥
2
c
s
n
log
(
n
s
+
1
)
r_3(s,n) \geq 2^{c \, sn \, \log (\frac {n}{s}+1)}
\]
for all
4
≤
s
≤
n
4 \leq s \leq n
. For constant
s
s
, this gives the first superexponential lower bound for
r
3
(
s
,
n
)
r_3(s,n)
, answering an open question posed by Erdős and Hajnal in 1972. Next, we consider the
3
3
-color Ramsey number
r
3
(
n
,
n
,
n
)
r_3(n,n,n)
, which is the minimum
N
N
such that every
3
3
-coloring of the triples of an
N
N
-element set contains a monochromatic set of size
n
n
. Improving another old result of Erdős and Hajnal, we show that
\[
r
3
(
n
,
n
,
n
)
≥
2
n
c
log
n
.
r_3(n,n,n) \geq 2^{n^{c \log n}}.
\]
Finally, we make some progress on related hypergraph Ramsey-type problems.
