On the Structure of the Set of Panchromatic Colorings of a Random Hypergraph

Abstract

The paper deals with the structure of the set of panchromatic 3-colorings of a random hypergraph in the uniform model $H(n,k,m)$. It is well known that panchromatic colorability with a given number of colors r has a sharp threshold, i.e., there exists a threshold value ${{\hat {m}}_{r}} = {{\hat {m}}_{r}}(n)$ such that, for any $\varepsilon > 0$, if $m\;\leqslant \;(1 - \varepsilon ){{\hat {m}}_{r}}$, then the random hypergraph $H(n,k,m)$ admits such a coloring with probability tending to 1 as $n \to \infty$, but, if $m\; \geqslant \;(1 + \varepsilon ){{\hat {m}}_{r}}$, then, vice versa, it does not admit such a coloring with probability tending to 1. We study the algorithmic threshold for panchromatic colorability with three colors and prove that, if the parameter m is slightly less than ${{\widehat m}_{3}}$, then the set of panchromatic 3-colorings of $H(n,k,m)$, though nonempty with probability tending to 1, undergoes shattering, which was first described by D. Achlioptas and A. Coja-Oghlan in 2008.