A word on $q$ symbols is a sequence of letters from a fixed alphabet of size $q$. For an integer $k\ge 1$, we say that a word $w$ is $k$-universal if, given an arbitrary word of length $k$, one can obtain it by removing entries from $w$. It is easily seen that the minimum length of a $k$-universal word on $q$ symbols is exactly $qk$. We prove that almost every word of size $(1+o(1))c_qk$ is $k$-universal with high probability, where $c_q$ is an explicit constant whose value is roughly $q\log q$. Moreover, we show that the $k$-universality property for uniformly chosen words exhibits a sharp threshold. Finally, by extending techniques of Alon [Geometric and Functional Analysis 27 (2017), no. 1, 1--32], we give asymptotically tight bounds for every higher dimensional analogue of this problem.
An array is row-Latin if no symbol is repeated within any row. An array is Latin if it and its transpose are both row-Latin. A transversal in an $n\times n$ array is a selection of $n$ different symbols from different rows and different columns. We prove that every $n \times n$ Latin array containing at least $(2-\sqrt{2}) n^2$ distinct symbols has a transversal. Also, every $n \times n$ row-Latin array containing at least $\frac14(5-\sqrt{5})n^2$ distinct symbols has a transversal. Finally, we show by computation that every Latin array of order $7$ has a transversal, and we describe all small…
A Latin array is a matrix of symbols in which no symbol occurs more than once within a row or within a column. A diagonal of an $n\times n$ array is a selection of $n$ cells taken from different rows and columns of the array. The weight of a diagonal is the number of different symbols on it. We show via computation that every Latin array of order $n\le11$ has a diagonal of weight at least $n-1$. A corollary is the existence of near transversals in Latin squares of these orders. More generally, for all $k\le20$ we compute a lower bound on the order of any Latin array that does not have a diagon…
Let m and n be integers, $2 \leq m \leq n$. An m by n array consists of mn cells, arranged in m rows and n columns, and each cell contains exactly one symbol. A transversal of an array consists of m cells, one from each row and no two from the same column. A latin transversal is a transversal in which no symbol appears more than once. We will establish a sufficient condition that a 3 by n array has a latin transversal.
A covering array $t$-$CA(n,k,g)$, of size $n$, strength $t$, degree $k$, and order $g$, is a $k\times n$ array on $g$ symbols such that every $t\times n$ sub-array contains every $t\times 1$ column on $g$ symbols at least once. Covering arrays have been studied for their applications to software testing, hardware testing, drug screening, and in areas where interactions of multiple parameters are to be tested. In this paper, we present an algebraic construction that improves many of the best known upper bounds on $n$ for covering arrays 4-$CA(n,k,g)$ with $g=3$. The $coverage$ $measure$ $μ_t(A)…
A covering array CA(N; t; k; v) is an N x k array on v symbols such that every N x t subarray contains as a row each t-tuple over the v symbols at least once. The minimum N for which a CA(N; t; k; v) exists is called the covering array number of t, k, and v, and it is denoted by CAN(t; k; v). In this work we prove new CANs using an optimized procedure.
A covering array $\rm{CA}(N;t,k,v)$ of strength $t$ is an $N \times k$ array of symbols from an alphabet of size $v$ such that in every $N \times t$ subarray, every $t$-tuple occurs in at least one row. A covering array is \emph{optimal} if it has the smallest possible $N$ for given $t$, $k$, and $v$, and \emph{uniform} if every symbol occurs $\lfloor N/v \rfloor$ or $\lceil N/v \rceil$ times in every column. Prior to this paper the only known optimal covering arrays for $t=2$ were orthogonal arrays, covering arrays with $v=2$ constructed from Sperner's Theorem and the Erdős-Ko-Rado Theorem, a…
Akbari and Alipour conjectured that any Latin array of order $n$ with at least $n^2/2$ symbols contains a transversal. We confirm this conjecture for large $n$, and moreover, we show that $n^{399/200}$ symbols suffice.
We consider the problem of constructing Latin cubes subject to the condition that some symbols may not appear in certain cells. We prove that there is a constant $γ> 0$ such that if $n=2^k$ and $A$ is $3$-dimensional $n\times n\times n$ array where every cell contains at most $γn$ symbols, and every symbol occurs at most $γn$ times in every line of $A$, then $A$ is {\em avoidable}; that is, there is a Latin cube $L$ of order $n$ such that for every $1\leq i,j,k\leq n$, the symbol in position $(i,j,k)$ of $L$ does not appear in the corresponding cell of $A$.
An $n \times n$ partial Latin square $P$ is called $α$-dense if each row and column has at most $αn$ non-empty cells and each symbol occurs at most $αn$ times in $P$. An $n \times n$ array $A$ where each cell contains a subset of $\{1,\dots, n\}$ is a $(βn, βn, βn)$-array if each symbol occurs at most $βn$ times in each row and column and each cell contains a set of size at most $βn$. Combining the notions of completing partial Latin squares and avoiding arrays, we prove that there are constants $α, β> 0$ such that, for every positive integer $n$, if $P$ is an $α$-dense $n \times n$ partial La…
Improved bounds on completion of partial Latin squares
Institut für Mathematik, Freie Universität Berlin（柏林自由大学数学研究所）；School of Mathematics, University of Birmingham（伯明翰大学数学学院）；School of Mathematics and Statistics, University of New South Wales（新南威尔士大学数学与统计学院）；Institute of Science and Technology Austria (ISTA)（奥地利科学与技术研究所）
We consider the problem of constructing Latin cubes subject to the condition that some symbols may not appear in certain cells. We prove that there is a constant $γ> 0$ such that if $n=2t$ and $A$ is a $3$-dimensional $n\times n\times n$ array where every cell contains at most $γn$ symbols, and every symbol occurs at most $γn$ times in every line of $A$, then $A$ is {\em avoidable}; that is, there is a Latin cube $L$ of order $n$ such that for every $1\leq i,j,k\leq n$, the symbol in position $(i,j,k)$ of $L$ does not appear in the corresponding cell of $A$.
Let $\textbf{T}(n,k)$ be the set of strings of length $n$ over the alphabet $Σ=\{1,2,\ldots,k\}$. A universal cycle for $\textbf{T}(n,k)$ can be constructed using a greedy algorithm: start with the string $k^n$, and continually append the least symbol possible without repeating a substring of length $n$. This construction also creates universal cycles for some subsets $\textbf{S}\subseteq\textbf{T}(n,k)$; we will classify all such subsets that are closed under rotations.
A latin hypercuboid of order $N$ is an $N\times...\times N\times k$ array filled with symbols from the set $\{0,...,N-1\} $ in such a way that every symbol occurs at most once in every line. If $k=N$, such an array is a latin hypercube. We prove that any latin hypercuboid of order 4 is completable to a latin hypercube. Keywords: latin hypercube, n-ary quasigroup
