Alexander Sidorenko

What we know and what we do not know about Turán numbers

The numbers which are traditionally named in honor of Paul Turán were introduced by him as a generalization of a problem he solved in 1941. The general problem of Turán having anextremely simple formulation but beingextremely hard to solve, has become one of the most fascinatingextremal problems in combinatorics. We describe the present situation and list conjectures which are not so hopeless.

A correlation inequality for bipartite graphs

We conjecture an integral inequality for a product of functionsh(xi,yj) where the diagram of the product is a bipartite graphG. In particular, this inequality states that the random graph with fixed numbers of vertices and edges contains the asymptotically minimal number of copies ofG.

Extremal Problems for Sets Forming Boolean Algebras and Complete Partite Hypergraphs

Three classes of finite structures are related by extremal properties: complete d-partite d-uniform hypergraphs, d-dimensional affine cubes of integers, and families of 2d sets forming a d-dimensional Boolean algebra. We review extremal results for each of these classes and derive new ones for Boolean algebras and hypergraphs, several obtained by employing relationships between the three classes. Related partition or coloring problems are also studied for Boolean algebras. Density results are given for Boolean algebras of sets all of whose atoms are the same size.

Upper Bounds for Turán Numbers

A system ofr-element subsets (blocks) of ann-element setXnis called aTuran(n, k, r)-systemif everyk-element subset ofXncontains at least one of the blocks. TheTuran number T(n, k, r) is the minimum size of such a system. We prove upper estimates:formula]

Spectral estimation of plasma fluctuations. I. Comparison of methods

The relative root mean squared errors (RMSE) of nonparametric methods for spectral estimation is compared for microwave scattering data of plasma fluctuations. These methods reduce the variance of the periodogram estimate by averaging the spectrum over a frequency bandwidth. As the bandwidth increases, the variance decreases, but the bias error increases. The plasma spectra vary by over four orders of magnitude, and therefore, using a spectral window is necessary. The smoothed tapered periodogram is compared with the adaptive multiple taper methods and hybrid methods. It is found that a hybrid method, which uses four orthogonal tapers and then applies a kernel smoother, performs best. For 300 point data segments, even an optimized smoothed tapered periodogram has a 24% larger relative RMSE than the hybrid method. Two new adaptive multitaper weightings which outperform Thomson’s original adaptive weighting are presented.

Boundedness of optimal matrices in extremal multigraph and digraph problems

We consider multigraphs in which any two vertices are joined by at mostq edges, and study the Turán-type problem for a given family of forbidden multigraphs. In the caseq=2, answering a question of Brown, Erdős and Simonovits, we obtain an explicit upper bound on the size of the matrix generating an asymptotical solution of the problem. In the caseq>2 we show that some analogous statements do not hold, and so disprove a conjecture of Brown, Erdős and Simonovits.

On the jumping constant conjecture for multigraphs

Abstract Let G be an infinite family of graphs closed under taking subgraphs. For each n, set e(n,G) = max {|E(G)|: G∈G,|V(G)| = n} Let density τ( G ) of a family G be defined by τ(G) = lim n→∞ e(n,G) n 2 The set of all possible densities of graph families was determined by Erdos and Stone in 1946. The analogous problem for multigraphs with multiplicity of edges not exceeding q (where q = 2, 3, …) appears to be much harder. In 1973 Brown, Erdos, and Simonovits conjectured that the set of all possible densities is well-ordered. They verified their conjecture for q = 2. If q = 1, this follows from the Erdos-Stone theorem. We disprove this conjecture for all q ≥ 4.

Function estimation using data-adaptive kernel smoothers—how much smoothing?

AbstractThe following sections are included:Bias-Versus-Variance Trade-offLocal Error and Optimal KernelsHow to Select the HalfwidthPlug-in-Derivative Estimates of the Local HalfwidthData-adaptive SmoothingFurther ReadingAcknowledgmentsReferences

Geometrical Techniques for Estimating Numbers of Linear Extensions

LetPbe a two-dimensional order, and __Pany complement ofP, i.e., any partial order whose comparability graph is the complement of the comparability graph ofP. Lete(Q) denote the number of linear extensions of the partial orderQ. Sidorenko showed thate(P)e(__P) ?n!, for any two-dimensional partial orderP. In this note, we use results from polyhedral combinatorics, and from the geometry ofRn, to give a companion upper bound one(P)e(__P), as well as an alternative proof of the lower bound. We use these results to obtain bounds on the number of linear extensions of a random two-dimensional partial order.

A partially ordered set of functionals corresponding to graphs

Abstract For a graph G whose vertices are v1,v2,…,vm and where E is the set of edges, we define a functional U G (h)=ʃʃ…ʃ ∏ {v i ,v j }∈E h(x i ,x j ) dμ(x 1 )dμ(x 2 )…dμ(x m ) , where h is a nonnegative symmetric function of two variables. We consider a binary relation ≽ for graphs with fixed numbers of vertices and edges, where G≽L means that UG(h)⩾UL(h) for every h. We prove that this relation is equivalent to the condition: the number of homomorphisms into every graph H from G is not less than from L. We obtain comparability and incomparability criteria and investigate the poset of k-edge trees. In particular, the first and the second maximal elements of this poset are found.