"Integer-making" theorems

Abstract

Let X = {x1, x2,...} be a finite set and associate to every xi a real number αi. Let f(n) [g (n)] be the least value such that given any family F of subsets of X having maximum degree n [cardinality n], one can find integers αi, i=1,2,... so that αi - αi|<1 and ∑ xi ε{lunate} Eai- ∑ xi ε{lunate} Eαi≤f{hook}(n) ∑ xi ε{lunate} Eai- ∑ xi ε{lunate} Eαi≤g(n) for all E ε{lunate} F. We prove f(n)≤n - 1 and g(n)≤c(n log n)1 2. © 1981.