We present recent results regarding rainbow matchings in bipartite graphs. Using topological methods we address a known conjecture of Stein and show that if Kn,n is partitioned into n sets of size n, then a partial rainbow matching of size 2n/3 exists. We generalize a result of Cameron and Wanless and show that for any n matchings of size n in a bipartite graph with 2n vertices there exists a f...

\Semantic uni cation" is the process of generating a basis set of substitutions (of terms for variables) that makes two given terms equal in a speci ed theory. Semantic uni cation is an important component of some theorem provers. \Semantic matching," a simpler variant of uni cation, where the substitution is made in only one of the terms, has potential usage in programming language interpreter...

König's theorem states that the covering number and the matching number of a bipartite graph are equal. We prove a generalisation, in which the point in one fixed side of the graph of each edge is replaced by a subtree of a given tree. The proof uses a recent extension of Hall's theorem to families of hypergraphs, by the first author and P. Haxell [3]. 1. Width and matching width of families of...

We characterize the bipartite stable b-matching polytope in terms of linear constraints. The stable b-matching polytope is the convex hull of the characteristic vectors of stable b-matchings, that is, of stable assignments of a two-sided multiple partner matching model. Our proof uses the comparability theorem of Roth and Sotomayor [13] and follows a similar line as Rothblum did in [14] for the...

We present a short and elementary proof of the Ajtai–Komlós–Tusnády (AKT) optimal matching theorem in dimension 2 via Fourier analysis smoothing argument. The upper bound applies to more general families samples, including dependent variables, interest study rates convergence for empirical measures. Following recent pde approach by L. Ambrosio, F. Stra D. Trevisan, we also adapt simple lower bo...

We characterize the completely determined Borel subsets of HYP as exactly [Formula: see text] HYP. As a result, believes there is well-ordering reals, that Dual Ramsey Theorem fails, and every d-regular bipartite graph has perfect matching, among other examples. Therefore, several theorems descriptive combinatorics are not theories hyperarithmetic analysis. In case Theorem, this answers questio...

Faudree, R.J., R.J. Gould and L.M. Lesniak, Neighborhood conditions and edge-disjoint perfect matchings, Discrete Mathematics 91 (1991) 33-43. A graph G satisfies the neighborhood condition ANC(G) 2 m if, for all pairs of vertices of G, the union of their neighborhoods has at least m vertices. For a fixed positive integer k, let G be a graph of even order n which satisfies the following conditi...

The matching preclusion number of a graph is the minimum number of edges whose deletion results in a graph that has neither perfect matchings nor almost-perfect matchings. For many interconnection networks, the optimal sets are precisely those induced by a single vertex. Recently, the conditional matching preclusion number of a graph was introduced to look for obstruction sets beyond those inci...