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An n x m sonar sequence is a subset of the n x m grid with exactly one point in each column, such that the (r~) vectors determined by them are all distinct. We show that for fixed n the maximal m for which a sonar sequence exists satisfies n-Cn 11/20 < m < n + 4n 2/3 for all n and m > n + c log n log log n for infinitely many n. Another problem concerns the(More)
1. The fall in chlorine concentration of Dakin's hypochlorite solution is more rapid in contact with necrotic than in contact with normal tissue. 2. The fall in chlorine concentration of chloramine-T solution is very slight when applied to necrotic tissue and is negligible when applied to normal tissue. 3. The action of the hypochlorite solution on tissue(More)
Here we show that Ramsey numbers M(k 1 , • • • , k..) give sharp upper bounds for the independence numbers of product graphs, in terms of the independence numbers of the factors. The Ramsey number M(k 1f • • • , kn) is the smallest integer m with the property that no matter how the (2) edges of the complete graph on m nodes are partitioned into n colors,(More)