SCHWEITZER 1976

Assume that $R$ , a recursive, binary relation on $\mathbb{N}$ (the set of natural numbers), orders $\mathbb{N}$ into type $\omega$ . Show that if $f(n)$ is the $n$ th element of this order, then $f$ is not necessarily recursive.

L. Posa

Let $G$ be an infinite graph such that for any countably infinite vertex set $A$ there is a vertex $p$ , not in $A$ , joined to infinitely many elements of $A$ . Show that $G$ has a countably infinite vertex set $A$ such that $G$ contains uncountably infinitely many vertices $p$ joined to infinitely many elements of $A$ .

P. Erdos, A. Hajnal

Let $H$ denote the set of those natural numbers for which $\tau(n)$ divides $n$ , where $\tau(n)$ is the number of divisors of $n$ . Show that

a) $n! \in H$ for all sufficiently large $n$ ,

b) $H$ has density $0$ .

P. Erdos

Let $\mathbb{Z}$ be the ring of rational integers. Construct an integral domain $I$ satisfying the following conditions:

a) $\mathbb{Z} \varsubsetneqq I$ ;

b) no element of $I - \mathbb{Z}$ (only in $I$ ) is algebraic over $\mathbb{Z}$ (that is, not a root of a polynomial with coefficients in $\mathbb{Z}$ );

c) $I$ only has trivial endomorphisms.

E. Fried

Let $S_{\nu}=\sum_{j=1}^n b_jz_j^{\nu} \;(\nu=0,\pm 1, \pm 2 ,...)$ , where the $b_j$ are arbitrary and the $z_j$ are nonzero complex numbers . Prove that $$|S_0| \leq n \max_{0<|\nu| \leq n} |S_{\nu}|.$$

G. Halasz

Let $0 \leq c \leq 1$ , and let $\eta$ denote the order type of the set of rational numbers. Assume that with every rational number $r$ we associate a Lebesgue-measurable subset $H_r$ of measure $c$ of the interval $[0,1]$ . Prove the existence of a Lebesgue-measurable set $H \subset [0,1]$ of measure $c$ such that for every $x \in H$ the set $$\{r : x \in H_r\ \}$$ contains a subset of type $\eta$ .

M. Laczkovich

Let $f_1,f_2,\dots,f_n$ be regular functions on a domain of the complex plane, linearly independent over the complex field. Prove that the functions $f_i\overline{f}_k, 1 \leq i,k \leq n$ , are also linearly independent.

L. Lempert

Prove that the set of all linearly combinations (with real coefficients) of the system of polynomials $\{ x^n+x^{n^2} \}_{n=0}^{\infty}$ is dense in $C[0,1]$ .

J. Szabados

Let $D$ be a convex subset of the $n$ -dimensional space, and suppose that $D'$ is obtained from $D$ by applying a positive central dilatation and then a translation. Suppose also that the sum of the volumes of $D$ and $D'$ is $1$ , and $D \cap D'\not= \emptyset .$ Determine the supremum of the volume of the convex hull of $D \cup D'$ taken for all such pairs of sets $D,D'$ .

L. Fejes-Toth, E. Makai