Rainbow Saturation for Complete Graphs

Abstract

We consider a supercritical Galton-Watson process Z n whose offspring distribution has mean m > 1 and is bounded by some d ∈ {2, 3, . . .}. As is well-known, the associated martingale W n = Z n /m n converges a.s. to some nonnegative random variable W ∞ . We provide a universal upper bound for the right tail of W ∞ and W n , which is uniform in n and in all offspring distributions with given m and d, namely:for some explicit constants c 1 , c 2 > 0. For a given offspring distribution, our upper bound decays exponentially as x → ∞, which is actually suboptimal, but our bound is universal : it provides a single effective expression -which does not require large x -and is valid simultaneously for all supercritical bounded offspring distributions.