This book is devoted to the study of classical combinatorial structures such as random graphs, permutations, and systems of random linear equations in finite fields. The author shows how the application of the generalized scheme of allocation in the study of random graphs and permutations reduces the combinatorial problems to classical problems of probability theory on the summation of independent random variables. He concentrates on research by Russian mathematicians, including a discussion of equations containing an unknown permutation and a presentation of techniques for solving systems of random linear equations in finite fields. These results will interest specialists in combinatorics and probability theory and will also be useful in applied areas of probabilistic combinatorics such as communication theory, cryptology, and mathematical genetics.

We introduce the notion of a hypercycle in a hypergraph which is defined by a Τ χ Ν (0, l)-matrix A. The maximum number of independent hypercycles s(A) is connected with the rank r (A) of the matrix A by the equality r(A) + s(A) -T. We prove that for a regular random hypergraph with N vertices and T edges, whose each edge contains not more than r vertices, the total number of hypercycles S(A) = 2*( Λ ) -1 has the following threshold property as Ν, Τ -» oo, Ν/Τ -* α: there exists a constant a r such that MS(A) -> 0 for α < a r , and MS(A) -» oo for a > a r . I. MAIN RESULTSWe consider a hypergraph G A which is defined by a Τ χ Ν matrix A = \\a tj \\ in GF (2). The set of vertices of the hypergraph G A is the set {1,..., TV} of the numbers of columns of the matrix, and the set of enumerated hyperedges is the set {61,..., &r}, where bt = {j: a tj = 1}, t = Ι,.,.,Τ. Thus, there exists a correspondence between a row a t = (a t i, -· -, CUN) and the hyperedge b t , t = 1,..., T. Note that the empty edge corresponds to a row consisting of zeros. The multiplicity of a vertex j in the set of hyperedges B = {6^,...,b tm } is the number of hyperedges in B which contain this vertex. A set of hyperedges B = {b tl ,..., b tm } is a hypercycle if every vertex of the hypergraph G A has an even multiplicity in B, in other words, if the sum by coordinates of rows a tl + ... + a tm in GF(2) is equal to the zero vector.If every row of the matrix A consists of exactly two units, then the hypergraph G A is an ordinary graph (parallel edges may be encountered), and a hypercycle is an ordinary cycle or a union of cycles.The set of numbers of hyperedges which form a hypercycle is called in [1] the critical set of rows of the matrix A.The union of two sets of hyperedges BI and B 2 is the set BI Δ Β 2 containing those, and only those, hyperedges which are contained in one of the sets BI and B 2 and are not contained in the both sets.Let ει,...,ε s take values 0 and 1. Hypercycles £?i,..., £ s are independent, if ειΒιΔε 2 Β 2 Δ...Δε 3 Β 3 = 0 iff ει = ... = ε β = 0. We denote by s(A) the maximum number of independent hypercycles in the hypergraph GA, and by r(A), the rank of the matrix A. In Section 2 it will be proved that r(A) + s(A) = T.(1)

For a T x n matrix A = )IaLjll in GF(2) we define a hypergraph G, with n vertices and T hyperedges e, = { j : a , = l}, t = 1, . . . , T . Denote a, = (afl, . . . , a,,), t = 1, . . . , T . A set of row numbers { t l , . . . , t,} is called a critical set if the sum of vectors a f l + . . . + a,_ is the zero vector. In terms of hypergraph G, a critical set can be interpreted as a hypercycle. We can naturally definite the concept of independence for critical sets. Let s ( A ) be the maximal number of independent critical sets in A . The rank r(A) of the matrix A and s ( A ) are connected by the equality r(A) + s ( A ) = T. The total number of critical sets S(A) is equal to FA) -1.Consider the following system of T random equations in GF(2):where i , ( t ) , . . . , i,(t), t = 1, . . . , T are independent identically distributed random variables which take values 1, . . . , n with equal probabilities. Denote A,,n,T the matrix of this system. We prove that the number S(A,,n,T) of critical sets in A r , n , T or hypercycles in GA,,,,T has a threshold property. Let n, T+ and Tln + a. Then for any fixed integer r z 3 there exists a constant a, such that MS(A,,n,T)+O if a

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