We prove several results from different areas of extremal combinatorics, including complete or partial solutions to a number of open problems. These results, coming mainly from extremal graph theory and Ramsey theory, have been collected together because in each case the relevant proofs are quite short.

Erdős and Rothschild [5] proposed a variant of the classical Turán problem in which they asked for the maximum number of edge-colorings of an n-vertex graph avoiding monochromatic copies of some given forbidden subgraph. The same variant has been addressed in [11] in connection with the Erdős–Ko–Rado Theorem for families of `-intersecting r-sets, which resulted in a fairly complete characteriza...

Let dα be a measure on R and let σ = (m1,m2, ..., mn), where mk ≥ 1, k = 1, 2, ..., n, are arbitrary real numbers. A polynomial ωn(x) := (x − x1)(x − x2)...(x − xn) with x1 ≤ x2 ≤ ... ≤ xn is said to be the n-th σ-orthogonal polynomial with respect to dα if the vector of zeros (x1, x2, ..., xn) is a solution of the extremal problem ∫

AMS 2000 subject classifications: 62H20 62P05 Keywords: Tail dependence Heavy tails Copula Multivariate extreme value distribution Marshall–Olkin distribution Archimedean copula Contagion risk a b s t r a c t The orthant tail dependence describes the relative deviation of upper-(or lower-) orthant tail probabilities of a random vector from similar orthant tail probabilities of a subset of its c...

We consider the stationary solutions of N = 4 supergravity coupled to n vector multiplets that define linear superpositions of non-interacting extremal black holes. The most general solutions of this type are derived from the graded decompositions of so(8, 2 + n) associated to its nilpotent orbits. We illustrate the formalism by giving explicitly asymptotically Minkowski non-BPS solutions of th...

Abstract: In this paper we study second order sufficient conditions for the strong-local optimality of singular Pontryagin extremals. In particular, we focus on the minimum-time problem for a control-affine system with vector inputs. We use Hamiltonian methods to prove that the coercivity of a suitably-defined second variation plus an involutivity assumption on the distribution of the controlle...

We study cyclic covering morphisms from M0,n to moduli spaces of unpointed stable curves of positive genus or compactified moduli spaces of principally polarized abelian varieties. Our main application is a construction of new semipositive vector bundles and nef divisors on M0,n, with a view toward the F-conjecture. In particular, we construct new extremal rays of Nef(M0,n/Sn). We also find an ...

The multivariate extremal index function relates the asymptotic distribution of the vector of pointwise maxima of a multivariate stationary sequence to that of the independent sequence from the same stationary distribution. It also measures the degree of clustering of extremes in the multivariate process. In this paper, we construct nonparametric estimators of this function and prove their asym...

Let G be a connected graph and Q(G) the signless Laplacian of G. The principal ratio γ(G) is maximum minimum entries Perron vector Q(G). In this paper, we consider among all graphs order n, show that for sufficiently large n extremal kite obtained by identifying an end vertex path to any complete graph.