Let and be sets of functions from domain X to ℝ. We say that validly generalises from approximate interpolation if and only if for each η > 0 and ∈, δ ∈ (0,1) there is m0(η, ∈, δ) such that for any function t ∈ and any probability distribution on X, if m > m0 then with m-probability at least 1 – δ, a sample X = (x1, X2,…,xm) ∈ Xm satisfies

We find conditions that are necessary and sufficient for to validly generalise from approximate interpolation, and we obtain bounds on the sample length m0{η,∈,δ) in terms of various parameters describing the expressive power of .

Let σ, π be two permutations selected at random from the uniform distribution on the symmetric group Sn. By a result of Dixon [5], the subgroup G generated by σ, π is almost always (i.e. with probability approaching 1 as n → ∞) either Sn or the alternating group An. We prove that the diameter of the Cayley graph of G defined by {σ, π} is almost always not greater than exp ((½ + o(l)). (In n)2).

We prove that a bipartite uniquely Hamiltonian graph has a vertex of degree 2 in each color class. As consequences, every bipartite Hamiltonian graph of minimum degree d has at least 21−dd! Hamiltonian cycles, and every bipartite Hamiltonian graph of minimum degree at least 4 and girth g has at least (3/2)g/8 Hamiltonian cycles. We indicate how the existence of more than one Hamiltonian cycle may lead to a general reduction method for Hamiltonian graphs.

A simple proof is given of the best-known upper bound on the cardinality of a set of vectors of length t over an alphabet of size b, with the property that, for every subset of k vectors, there is a coordinate in which they all differ. This question is motivated by the study of perfect hash functions.

Béla Bollobás [1] conjectured the following. For any positive integer Δ and real 0 < c < ½ there exists an n0 with the following properties. If n ≥ n0, T is a tree of order n and maximum degree Δ, and G is a graph of order n and maximum degree not exceeding cn, then there is a packing of T and G. Here we prove this conjecture. Auxiliary Theorem 2.1 is of independent interest.

This note contains a refinement of our paper [8], leading to an alternative proof of a conjecture of Mader and of Erdős and Hajnal recently proved by Bollobás and Thomason.

Suppose X1, X2,… is a sequence of independent and identically distributed random elements whose values are taken in a finite set S of size |S| ≥ 2 with probability distribution ℙ(X = s) = p(s) > 0 for s ∈ S. Pevzner has conjectured that for every probability distribution ℙ there exists an N > 0 such that for every word A with letters in S whose length is at least N, there exists a second word B of the same length as A, such that the event that B appears before A in the sequence X1, X2,… has greater probability than that of A appearing before B. In this paper it is shown that a distribution ℙ satisfies Pevzner's conclusion if and only if the maximum value of ℙ, p, and the secondary maximum c satisfy the inequality . For |S| = 2 or |S| = 3, the inequality is true and the conjecture holds. If , then the conjecture is true when A is not allowed to consist of pure repetitions of that unique element for which the distribution takes on its mode.

A linear forest is the union of a set of vertex disjoint paths. Akiyama, Exoo and Harary, and independently Hilton, have conjectured that the edges of every graph of maximum degree Δ can be covered by linear forests. We show that almost every graph can be covered with this number of linear forests.

The smallest minimal degree of an r-partite graph that guarantees the existence of a complete subgraph of order r has been found for the case r = 3 by Bollobás, Erdő and Szemerédi, who also gave bounds for the cases r ≥ 4. In this paper the exact value is established for the cases r = 4 and 5, and the bounds for r ≥ 6 are improved.

It is shown that, for every integer v < 7, there is a connected graph in which some v longest paths have empty intersection, but any v – 1 longest paths have a vertex in common. Moreover, connected graphs having seven or five minimal sets of longest paths (longest cycles) with empty intersection are presented. A 26-vertex 2-connected graph whose longest paths have empty intersection is exhibited.

Morphological analysis, which is at the heart of the processing of natural language requires computationally effective morphological processors. In this paper an approach to the organization of an inflectional morphological model and its application for the Russian language are described. The main objective of our morphological processor is not the classification of word constituents, but rather an efficient computational recognition of morpho-syntactic features of words and the generation of words according to requested morpho-syntactic features. Another major concern that the processor aims to address is the ease of extending the lexicon. The templated word-paradigm model used in the system has an engineering flavour: paradigm formation rules are of a bottom-up (word specific) nature rather than general observations about the language, and word formation units are segments of words rather than proper morphemes. This approach allows us to handle uniformly both general cases and exceptions, and requires extremely simple data structures and control mechanisms which can be easily implemented as a finite-state automata. The morphological processor described in this paper is fully implemented for a substantial subset of Russian (more then 1,500,000 word-tokens – 95,000 word paradigms) and provides an extensive list of morpho-syntactic features together with stress positions for words utilized in its lexicon. Special dictionary management tools were built for browsing, debugging and extension of the lexicon. The actual implementation was done in C and C++, and the system is available for the MS-DOS, MS-Windows and UNIX platforms.

A family ℱ of k-element sets of an n-set is called t-intersecting if any two of its members overlap in at least t-elements. The Erdős-Ko-Rado Theorem gives a best possible upper bound for such a family if n ≥ n0(k, t). One of the most exciting open cases is when t = 2, n = 2k. The present paper gives an essential improvement on the upper bound for this case. The proofs use linear algebra and yield more general results.

Let G be a minimally n-edge-connected finite simple graph with vertex number |G| ≥ 2n + 2 + [3/n] and let n ≥ 3 be odd. It is proved that the number of vertices of degree n in G is at least ((n − 1 − ∈n)/(2n + 1))|G| + 2 + 2∈n, where ∈n = (3n + 3)/(2n2 − 3n − 3), and that for every n ≡ 3 (mod 4) this lower bound is attained by infinitely many minimally n-edge-connected finite simple graphs.

The asymptotic distribution of the number of Hamilton cycles in a random regular graph is determined. The limit distribution is of an unusual type; it is the distribution of a variable whose logarithm can be written as an infinite linear combination of independent Poisson variables, and thus the logarithm has an infinitely divisible distribution with a certain discrete Lévy measure. Similar results are found for some related problems. These limit results imply that some different models of random regular graphs are contiguous, which means that they are qualitatively asymptotically equivalent. For example, if r > 3, then the usual (uniformly distributed) random r-regular graph is contiguous to the one constructed by taking the union of r perfect matchings on the same vertex set (assumed to be of even cardinality), conditioned on there being no multiple edges. Some consequences of contiguity for asymptotic distributions are discussed.

An element e of a matroid M is called non-binary when M\e and M/e are both non-binary matroids. Oxley in [5] gave a characterization of the 3-connected non-binary matroids without non-binary elements. In this paper, we will construct all the 3-connected matroids having 1, 2 or 3 non-binary elements.