On the longest run of coincidences

Abstract

Consider the rectangles of the first k(n) lines of length n in the right-upper integer-lattice, and suppose that its points are labelled randomly by the numbers 1, 2,..., m. The time i is called coincidence if the points (i, 1), (i, 2),..., (i, k(n)) are labelled identically. Asymptotic properties of the longest run of coincidences are discussed under different conditions on k(n). The results are related to a problem of P. Révész: If the points of an n ×n integer-lattice are coloured red and white randomly, what is the largest area of rectangles with red points only. A conjecture is formulated, indicating some peculiar number-theoretic characteristics of some limit-relations in this area. © 1982 Springer-Verlag.