Let K be the class of all inverse limits G=lim(<- n is an element of) G(n), where each Gn is a finite ordered graph. G is an element of K is universal if every B is an element of K embeds continuously into G. Theorem (1). For every finite ordered graph A there exists a least natural number k ( A) >= 1 such that for every universal G is an element of K, for every finite Baire measurable partition of the set ((G)(A)) of all copies of A in G, there is a closed copy G' subset of G of G such that ((G)(A)') meets atNGn meets at most k(A) parts. In the arrow notation: G -> Baire(G)<infinity|k(A). Theorem (2). The probability that k(A)=1, for a finite ordered graph A, chosen randomly with uniform probability from all graphs on \{0,1,...,n-1\}, tends to 1 as n grows to infinity, where k(A) is the number given by Theorem (1). As a corollary Theorem (3). The class K with Baire partitions satisfies with high probability the A-partition property for a finite ordered graph A, where the A-partition property is (for all B is an element of K)(there exists C is an element of K)C -> Baire(B)A.