SCHWEITZER 1986

If $(A, <)$ is a partially ordered set, its dimension, $\dim (A, <)$ , is the least cardinal $\kappa$ such that there exist $\kappa$ total orderings $\{ <_{\alpha} \colon \alpha < \kappa \}$ on $A$ with $<=\cap_{\alpha < \kappa} <_\alpha$ . Show that if $\dim (A, <)>\aleph_0$ , then there exist disjoint $A_0, A_1\subseteq A$ with $\dim (A_0, <)$ , $\dim (A_1, <)>\aleph_0$ . [D. Kelly, A. Hajnal, B. Weiss]

Show that if $k\leq \frac n2$ and $\mathcal F$ is a family $k\times k$ submatrices of an $n\times n$ matrix such that any two intersect then $$|\mathcal F|\leq \binom{n-1}{k-1}^2$$ [Gy. Katona]

(a) Prove that for every natural number $k$ , there are positive integers $a_1<a_2<\ldots <a_k$ such that $a_i-a_j$ divides $a_i$ for all $1\leq i, j\leq k, i\neq j$ .

(b) Show that there is an absolute constant $C>0$ such that $a_1>k^{Ck}$ for every sequence $a_1,\ldots, a_k$ of numbers that satisfy the above divisibility condition. [A. Balogh, I. Z. Ruzsa]

Determine all real numbers $x$ for which the following statement is true: the field $\mathbb C$ of complex numbers contains a proper subfield $F$ such that adjoining $x$ to $F$ we get $\mathbb C$ . [M. Laczkovich]

Prove that existence of a constant $c$ with the following property: for every composite integer $n$ , there exists a group whose order is divisible by $n$ and is less than $n^c$ , and that contains no element of order $n$ . [P. P. Palfy]

Let $U$ denote the set $\{ f\in C[0, 1] \colon |f(x)|\leq 1 \mathrm{for}\,\mathrm{all} x\in [0, 1]\}$ . Prove that there is no topology on $C[0, 1]$ that, together with the linear structure of $C[0,1]$ , makes $C[0,1]$ into a topological vector space in which the set $U$ is compact. (Assume that topological vector spaces are Hausdorff) [V. Totik]

Prove that the series $\sum_p c_p f(px)$ , where the summation is over all primes, unconditionally converges in $L^2[0,1]$ for every $1$ -periodic function $f$ whose restriction to $[0,1]$ is in $L^2[0,1]$ if and only if $\sum_p |c_p|<\infty$ . (Unconditional convergence means convergence for all rearrangements.) [G. Halasz]

Let $a_0=0$ , $a_1, \ldots, a_k$ and $b_1, \ldots, b_k$ be arbitrary real numbers. (i) Show that for all sufficiently large $n$ there exist polynomials $p_n$ of degree at most $n$ for which $$p_n^{(i)} (-1)=a_i,\,\,\,\, p_n^{(i)} (1)=b_i,\,\,\,\, i=0, 1, \ldots, k$$ and $$\max_{|x|\leq 1} |p_n (x)|\leq \frac{c}{n^2}\,\,\,\,\,\,\,\,\, (*)$$ where the constant $c$ depends only on the numbers $a_i, b_i$ .

(ii) Prove that, in general, (*) cannot be replaced by the relation $$\lim_{n\to\infty} n^2\cdot \max_{|x|\leq 1} |p_n (x)| = 0$$ [J. Szabados]

Consider a latticelike packing of translates of a convex region $K$ . Let $t$ be the area of the fundamental parallelogram of the lattice defining the packing, and let $t_{\min} (K)$ denote the minimal value of $t$ taken for all latticelike packings. Is there a natural number $N$ such that for any $n>N$ and for any $K$ different from a parallelogram, $nt_{\min} (K)$ is smaller that the area of any convex domain in which $n$ translates to $K$ can be placed without overlapping? (By a latticelike packing of $K$ we mean a set of nonoverlapping translates of $K$ obtained from $K$ by translations with all vectors of a lattice.) [G. and L. Fejes-Toth]

Let $X_1, X_2$ be independent, identically distributed random variables such that $X_i\geq 0$ for all $i$ . Let $\mathrm EX_i=m$ , $\mathrm{Var} (X_i)=\sigma ^2<\infty$ . Show that, for all $0<\alpha\leq 1$ $$\lim_{n\to\infty} n\,\mathrm{Var} \left( \left[ \frac{X_1+\ldots +X_n}{n}\right] ^\alpha\right)=\frac{\alpha ^ 2 \sigma ^ 2}{m^{2(1-\alpha)}}$$ [Gy. Michaletzki]