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A set of $N$ permutations of $\{1,2,\ldots,v\}$ is $t$-suitable, if eachsymbol precedes each subset of $t-1$ others in at least one permutation. Theextremal 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 Spencerin 1971. Colbourn recently introduced the concept of suitable cores asequivalent objects of suitable sets of permutations, and studied the dualproblem of determining the largest $v=\text{SCN}(t,N)$ such that a suitablecore 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 ofSCN$(t,N)$ is asymptotically $\lfloor\frac{t}{2}\rfloor+2$ if $l$ is a fixedinteger. In this paper, we improve this result by showing that it is also truewhen $l=O(\ln t)$ using Ramsey theory. When $v$ is bigger than$\lfloor\frac{t}{2}\rfloor+2$, we give new explicit constructions of suitablecores from packings of triples, and random constructions from extended Ramseycolorings.

Suitable sets of permutations, packings of triples, and Ramsey's theorem