  In this paper we study a family of discrete configuration spaces, the
so-called protocol complexes, which are of utmost importance in theoretical
distributed computing. Specifically, we consider questions of the existance of
compliant binary labelings on the vertices of iterated standard chromatic
subdivisions of an n-simplex. The existance of such labelings is equivalent to
the existance of distributed protocols solving Weak Symmetry Breaking task in
the standard computational model.
  As a part of our formal model, we introduce function sb(n), defined for
natural numbers n, called the symmetry breaking function. From the geometric
point of view sb(n) denotes the minimal number of iterations of the standard
chromatic subdivision of an (n-1)-simplex, which is needed for the compliant
binary labeling to exist. From the point of distributed computing, the function
sb(n) measures the minimal number of rounds in a protocol solving the Weak
Symmetry Breaking task.
  In addition to the development of combinatorial topology, which is applicable
in a broader context, our main contribution is the proof of new bounds for the
function sb(n). Accordingly, the bulk of the paper is taken up by in-depth
analysis of the structure of adjacency graph on the set of n-simplices in
iterated standard chromatic subdivision of an n-simplex. On the algorithmic
side, we provide the first distributed protocol solving Weak Symmetry Breaking
task in the layered immediate snapshot computational model for some number of
processes.
  It is well known, that the smallest number of processes for which Weak
Symmetry Breaking task is solvable is 6. Based on our analysis, we are able to
find a very fast explicit protocol, solving the Weak Symmetry Breaking for 6
processes using only 3 rounds. Furthermore, we show that no protocol can solve
Weak Symmetry Breaking in fewer than 2 rounds.
