Abstract A sequence covering array, denoted by ( N ; t , v ), is a set of N permutations of $$\{0, \dots , v-1 \}$$ { 0 , ⋯ , v - 1 } such that each sequence of t distinct elements of $$\{0, \dots , v-1\}$$ { 0 , ⋯ , v - 1 } is a (not necessarily contiguous) subsequence of at least one permutation. The minimum number of permutations such a sequence covering array can have is t ! and it has been conjectured that for $$t > 4$$ t > 4 , if a sequence covering array with t ! permutations exists, then $$v \in \{t,t+1\}$$ v ∈ { t , t + 1 } . In this paper, we prove that an (7!; 7, 10) does not exist. We do this by analysing connections between sequence covering arrays and a special kind of covering array called an excess coverage array.

The Lovász Local Lemma is a powerful probabilistic technique for proving the existence of combinatorial objects. It is especially useful for coloring graphs and hypergraphs with bounded maximum degree. This paper presents a general theorem for coloring hypergraphs that in many instances matches or slightly improves upon the bounds obtained using the Lovász Local Lemma. Moreover, the theorem directly shows that there are exponentially many colorings. The elementary and self-contained proof is inspired by a recent result for nonrepetitive colorings by Rosenfeld [ Electron. J. Combin., 27 (2020), P3.43]. We apply our general theorem in the settings of proper hypergraph coloring, proper graph coloring, independent transversals, star coloring, nonrepetitive coloring, frugal coloring, Ramsey number lower bounds, and $k$-SAT.

We define a cover of a Latin square to be a set of entries that includes at least one representative of each row, column and symbol. A cover is minimal if it does not contain any smaller cover. A partial transversal is a set of entries that includes at most one representative of each row, column and symbol. A partial transversal is maximal if it is not contained in any larger partial transversal. We explore the relationship between covers and partial transversals. We prove the following: (1) The minimum size of a cover in a Latin square of order n is $$n+a$$ if and only if the maximum size of a partial transversal is either $$n-2a$$ or $$n-2a+1$$ . (2) A minimal cover in a Latin square of order n has size at most $$\mu _n=3(n+1/2-\sqrt{n+1/4})$$ . (3) There are infinitely many orders n for which there exists a Latin square having a minimal cover of every size from n to $$\mu _n$$ . (4) Every Latin square of order n has a minimal cover of a size which is asymptotically equal to $$\mu _n$$ . (5) If $$1\leqslant k\leqslant n/2$$ and $$n\geqslant 5$$ then there is a Latin square of order n with a maximal partial transversal of size $$n-k$$ . (6) For any $$\varepsilon >0$$ , asymptotically almost all Latin squares have no maximal partial transversal of size less than $$n-n^{2/3+\varepsilon }$$ .

It has been conjectured that every Steiner triple system of order $v \neq 7$ has chromatic index at most $(v+3)/2$ when $v \equiv 3 {\:({\rm mod}\ 6)}$ and at most $(v+5)/2$ when $v \equiv 1 {\:({\rm mod}\ 6)}$. Herein, we construct a Steiner triple system of order $v$ with chromatic index at least $(v+3)/2$ for each integer $v \equiv 3 {\:({\rm mod}\ 6)}$ such that $v \geqslant 15$, with four possible exceptions. We further show that the maximum number of disjoint parallel classes in the systems constructed is sublinear in $v$. Finally, we establish for each order $v \equiv 15 {\:({\rm mod}\ 18)}$ that there are at least $v^{v^2(1/6+o(1))}$ nonisomorphic Steiner triple systems with chromatic index at least $(v+3)/2$ and that some of these systems are cyclic.

We show that for all integers $m \geqslant 4$ there exists a $2m\times 2m\times m$ latin cuboid that cannot be completed to a $2m\times 2m\times 2m$ latin cube. We also show that for all even $m>2$ there exists a $(2m{-}1)\times(2m{-}1)\times(m{-}1)$ latin cuboid that cannot be extended to any $(2m{-}1)\times(2m{-}1)\times m$ latin cuboid.

We count all latin cubes of order $n\le6$ and latin hypercubes of order $n\le5$ and dimension $d\le5$. We classify these (hyper)cubes into isotopy classes and paratopy classes (main classes). For the same values of n and d we classify all d-ary quasigroups of order n into isomorphism classes and also count them according to the number of identity elements they possess (meaning we have counted the d-ary loops). We also give an exact formula for the number of (isomorphism classes of) d-ary quasigroups of order 3 for every d. Then we give a number of constructions for d-ary quasigroups with a specific number of identity elements. In the process, we prove that no 3-ary loop of order n can have exactly $n-1$ identity elements (but no such result holds in dimensions other than 3). Finally, we give some new examples of latin cuboids which cannot be extended to latin cubes.