SCHWEITZER 1974

Let $\mathcal{F}$ be a family of subsets of a ground set $X$ such that $\cup_{F \in \mathcal{F}}F=X$ , and

(a) if $A,B \in \mathcal{F}$ , then $A \cup B \subseteq C$ for some $C \in \mathcal{F};$

(b) if $A_n \in \mathcal{F} \;(n=0,1,...)\ , B \in \mathcal{F},$ and $A_0 \subset A_1 \subset...,$ then, for some $k \geq 0, A_n \cap B=A_k \cap B$ for all $n \geq k$ .

Show that there exist pairwise disjoint sets ${ X_{\gamma} \;( \gamma \in \Gamma}\ )$ , with $X= \cup \{ X_{\gamma} : \;\gamma \in \Gamma \ \},$ such that every $X_{\gamma}$ is contained in some member of $\mathcal{F}$ , and every element of $\mathcal{F}$ is contained in the union of finitely many $X_{\gamma}$ 's.

A. Hajnal

Let $G$ be a $2$ -connected nonbipartite graph on $2n$ vertices. Show that the vertex set of $G$ can be split into two classes of $n$ elements such that the edges joining the two classes form a connected, spanning subgraph.

L. Lovasz

Prove that a necessary and sufficient for the existence of a set $S \subset \{1,2,...,n \}$ with the property that the integers $0,1,...,n-1$ all have an odd number of representations in the form $x-y, x,y \in S$ , is that $(2n-1)$ has a multiple of the form $2.4^k-1$

L. Lovasz, J. Pelikan

Let $R$ be an infinite ring such that every subring of $R$ different from $\{0 \}$ has a finite index in $R$ . (By the index of a subring, we mean the index of its additive group in the additive group of $R$ .) Prove that the additive group of $R$ is cyclic.

L. Lovasz, J. Pelikan

Let $\{f_n \}_{n=0}^{\infty}$ be a uniformly bounded sequence of real-valued measurable functions defined on $[0,1]$ satisfying $$\int_0^1 f_n^2=1.$$ Further, let $\{ c_n \}$ be a sequence of real numbers with $$\sum_{n=0}^{\infty} c_n^2= +\infty.$$ Prove that some re-arrangement of the series $\sum_{n=0}^{\infty} c_nf_n$ is divergent on a set of positive measure.

J. Komlos

Let $f(x)=\sum_{n=1}^{\infty} a_n/(x+n^2), \;(x \geq 0)\ ,$ where $\sum_{n=1}^{\infty} |a_n|n^{- \alpha} < \infty$ for some $\alpha > 2$ . Let us assume that for some $\beta > 1/{\alpha}$ , we have $f(x)=O(e^{-x^{\beta}})$ as $x \rightarrow \infty$ . Prove that $a_n$ is identically $0$ .

G. Halasz

Given a positive integer $m$ and $0 < \delta <\pi$ , construct a trigonometric polynomial $f(x)=a_0+ \sum_{n=1}^m (a_n \cos nx+b_n \sin nx)$ of degree $m$ such that $f(0)=1, \int_{ \delta \leq |x| \leq \pi} |f(x)|dx \leq c/m,$ and $\max_{-\pi \leq x \leq \pi}|f'(x)| \leq c/{\delta}$ , for some universal constant $c$ .

G. Halasz

Prove that there exists a topological space $T$ containing the real line as a subset, such that the Lebesgue-measurable functions, and only those, extend continuously over $T$ . Show that the real line cannot be an everywhere-dense subset of such a space $T$ .

A. Csaszar

Let $A$ be a closed and bounded set in the plane, and let $C$ denote the set of points at a unit distance from $A$ . Let $p \in C$ , and assume that the intersection of $A$ with the unit circle $K$ centered at $p$ can be covered by an arc shorter that a semicircle of $K$ . Prove that the intersection of $C$ with a suitable neighborhood of $p$ is a simple arc which $p$ is not an endpoint.

M. Bognar

Let $\mu$ and $\nu$ be two probability measures on the Borel sets of the plane. Prove that there are random variables $\xi_1, \xi_2, \eta_1, \eta_2$ such that

(a) the distribution of $(\xi_1, \xi_2)$ is $\mu$ and the distribution of $(\eta_1, \eta_2)$ is $\nu$ ,

(b) $\xi_1 \leq \eta_1, \xi_2 \leq \eta_2$ almost everywhere, if an only if $\mu(G) \geq \nu(G)$ for all sets of the form $G=\cup_{i=1}^k (-\infty, x_i) \times (-\infty, y_i).$

P. Major