Finding Patterns Avoiding Many Monochromatic Constellations

Abstract

Given fixed 0 = q 0 < q 1 < q 2 < … < qk = 1, a constellation in [n] is a scaled translated realization of the qi with all elements in [n], i.e., p, p + q 1 d, p + q 2 d, …, p + q k–1 d, p + d. We consider the problem of minimizing the number of monochromatic constellations in a two-coloring of [n]. We show how, given a coloring based on a block pattern, to find the number of monochromatic solutions to a lower-order term, and also how experimentally we might find an optimal block pattern. We also show for the case k = 2 that there is always a block pattern that beats random coloring.

Locations

• Experimental Mathematics - View
• Project Euclid (Cornell University) - View - PDF