  A set of $N$ permutations of $\{1,2,\ldots,v\}$ is $t$-suitable, if each
symbol precedes each subset of $t-1$ others in at least one permutation. The
extremal problem of determining the smallest size $N$ of such sets for given
$v$ and $t$ was the subject of classical studies by Dushnik in 1950 and Spencer
in 1971. Colbourn recently introduced the concept of suitable cores as
equivalent objects of suitable sets of permutations, and studied the dual
problem of determining the largest $v=\text{SCN}(t,N)$ such that a suitable
core exists for given $t$ and $N$. Chan and Jedwab showed that when
$N=\lfloor\frac{t+1}{2}\rfloor\lceil\frac{t+1}{2}\rceil+l$, the value of
SCN$(t,N)$ is asymptotically $\lfloor\frac{t}{2}\rfloor+2$ if $l$ is a fixed
integer. In this paper, we improve this result by showing that it is also true
when $l=O(\ln t)$ using Ramsey theory. When $v$ is bigger than
$\lfloor\frac{t}{2}\rfloor+2$, we give new explicit constructions of suitable
cores from packings of triples, and random constructions from extended Ramsey
colorings.
