  We provide several constructions for problems in Ramsey theory. First, we
prove a superexponential lower bound for the classical 4-uniform Ramsey number
$r_4(5,n)$, and the same for the iterated $(k-4)$-fold logarithm of the
$k$-uniform version $r_k(k+1,n)$. This is the first improvement of the original
exponential lower bound for $r_4(5,n)$ implicit in work of Erd\H os and Hajnal
from 1972 and also improves the current best known bounds for larger $k$ due to
the authors. Second, we prove an upper bound for the hypergraph Erd\H os-Rogers
function $f^k_{k+1, k+2}(N)$ that is an iterated $(k-13)$-fold logarithm in
$N$. This improves the previous upper bounds that were only logarithmic and
addresses a question of Dudek and the first author that was reiterated by
Conlon, Fox and Sudakov. Third, we generalize the results of Erd\H os and
Hajnal about the 3-uniform Ramsey number of $K_4$ minus an edge versus a clique
to $k$-uniform hypergraphs.
