We show that for any positive integer k 4, if R is a (2k − 1)× (2k − 1) partial Latin square, then R is avoidable given that R contains an empty row, thus extending a theorem of Chetwynd and Rhodes. We also present the idea of avoidability in the setting of partial r-multi Latin squares, and give some partial fillings which are avoidable. In particular, we show that ifR contains at most nr/2 sy...

A set S of positive integers is avoidable if there exists a partition of the positive integers into two disjoint sets such that no two distinct integers from the same set sum to an element of S. Much previous work has focused on proving the avoidability of very special sets of integers. We vastly broaden the class of avoidable sets by establishing a previously unnoticed connection with the elem...

Cassaigne conjectured in 1994 that any pattern with m distinct variables of length at least 3(2m−1) is avoidable over a binary alphabet, and any pattern with m distinct variables of length at least 2 is avoidable over a ternary alphabet. Building upon the work of Rampersad and the power series techniques of Bell and Goh, we obtain both of these suggested strict bounds. Similar bounds are also o...

Introduction This aim of this study is to narrow patients truly at risk for avoidable readmissions, excluding the ones who are identified as planned readmissions. The Belgian administrative dataset contains additional information not collected in the USA dataset, namely the ‘planned’ versus ‘unplanned’ admission status. Using this flag allows us to identify readmissions that are probably not av...

A Pythagorean triple is a triple of positive integers a,b,c ∈ N+ satisfying a2 + b2 = c2. Is it true that, for any finite coloring of N+, at least one Pythagorean triple must be monochromatic? In other words, is the Diophantine equation X2 +Y 2 = Z2 regular? This problem has been open since several decades, even restricted to 2-colorings. In this note, we introduce partial morphisms, which are ...

The strongest risk factors for avoidable hospital admission are age and deprivation but ethnicity, distance to hospital, rurality, lifestyle and meteorological factors are also important, as well as access to primary care. There is still considerable uncertainty around which admissions are avoidable. In terms of services to reduce admissions there is evidence of effectiveness for education, sel...

In combinatorics on words, a word w over an alphabet Σ is said to avoid a pattern p over an alphabet ∆ of variables if there is no factor f of w such that f = h(p) where h : ∆∗ → Σ∗ is a non-erasing morphism. A pattern p is said to be k-avoidable if there exists an infinite word over a k-letter alphabet that avoids p. We consider the patterns such that at most two variables appear at least twic...