Let G ⊂C be a bounded simply connected domain with boundary Γ and let E ⊂G be a regular compact set with connected complement. In this paper we investigate asymptotics of the extremal constants: χn = inf p∈Pkn sup q∈Pn−kn ||pq ||E ||pq ||Γ , n = 1,2, . . . , where ‖ ·‖K is the supremum norm on a compact set K ,Pm is the set of all algebraic polynomials of degree at most m, and kn/n→ θ ∈ [0,1] a...

We investigate the maximum number of edges in a graph with a prescribed number of 1factors. We also examine the structure of such extremal graphs.

A graph is said to be Ck-saturated if it contains no cycles of length k but does contain such a cycle after the addition of any edge in the complement of the graph. Determining the minimum size of Ck-saturated graphs is one of the interesting problems on extremal graphs. The exact minimum sizes are known for k = 3, 4 and 5, but only general bounds are shown for k ≥ 6. This paper deals with boun...

We derive bounds for f(v), the maximum number of edges in a graph on v vertices that contains neither three-cycles nor four-cycles. Also, we give the exact value of f(v) for all v up to 24 and constructive lower bounds for all v up to 200.

For an integer l ≥ 2, let U(l) be the class of matroids with no U2,l+2-minor. A matroid in U(l) is extremal if it is simple and has no simple rank-preserving singleelement extension in U(l). An amalgam of two matroids is a simultaneous extension of both on the union of the two ground sets. We study amalgams of extremal matroids in U(l): we determine which amalgams are in U(l) and which are extr...

A unique type of subcontexts is always present in formal contexts with many concepts: the contranominal scales. We make this precise by giving an upper bound for the number of minimal generators (and thereby for the number of concepts) of contexts without contranominal scales larger than a given size. Extremal contexts are constructed which meet this bound exactly. They are completely classified.

The main purpose of this paper is to prove the uniqueness of a graph attaining the maximum of the number of independent sets over all $k$-regular graphs on $n$ vertices for $2k|n$.