Extremal problems on probability distributions

On Multigraph Extremal Problems

1 . Introduction . In this paper we shall consider multigraphs and digraphs (= directed graphs) with bounded multiplicity : an integer r is fixed and we shall assume, that the considered multigraphs or digraphs have no loops, further, if u and v are two vertices of a multigraph M, they can be joined by more than one edge, however, they cannot be joined by more than r edges. In case of digraphs ...

Extremal problems on ∆-systems

A family of sets is called a ∆-system (respectively, a weak ∆-system) if the intersection of any two sets is the same (respectively, the cardinality of the intersection of any two sets is the same). In 1960, P.Erdős and R.Rado started studying the maximum size of a k-uniform hypergraph not containing a ∆-system of a given size. The aim of the present article is to survey the progress and state ...

Extremal problems on set systems

Bounds on marginal probability distributions

We propose a novel bound on single-variable marginal probability distributions in factor graphs with discrete variables. The bound is obtained by propagating local bounds (convex sets of probability distributions) over a subtree of the factor graph, rooted in the variable of interest. By construction, the method not only bounds the exact marginal probability distribution of a variable, but also...

On some extremal problems on r-graphs

Denote by G(r)(n ; k) an r-graph of n vertices and k r-tuples. TurLn’s classical problem states: Determine the smallest integer f(n ;r, I) so that every G(‘)(n ; f (n ; r, I)) contains a K(‘)(I). Turin determined f (n; r, I) for r = 2, but nothing is known for r > 2. Put lim,=., f (n ; r, l)/(y) = cr,l. The values of cr l are not known for r > 2. I prove that to every E > 0 and integer t there ...