SCHWEITZER 2008

Let $H \subset P(X)$ be a system of subsets of $X$ and $\kappa > 0$ be a cardinal number such that every $x \in X$ is contained in less than $\kappa$ members of $H$ . Prove that there exists an $f \colon X \rightarrow \kappa$ coloring, such that every nonempty $A \in H$ has a "unique" point, that is, an element $x \in A$ such that $f(x) \neq f(y)$ for all $x \neq y \in A$ .

(translated by Miklós Maróti)

Let $t\ge 3$ be an integer, and for $1\le i <j\le t$ let $A_{ij}=A_{ji}$ be an arbitrary subset of an $n$ -element set $X$ . Prove that there exist $1\le i < j\le t$ for which $$\left| \left( X\,\backslash A_{ij}\right) \cup \bigcup_{k\neq i,j}\left( A_{ik}\cap A_{jk}\right) \right| \ge \frac{t-2}{2t-2}n$$ (translated by Miklós Maróti)

A bipartite graph on the sets $\{ x_1,\ldots, x_n \}$ and $\{ y_1,\ldots, y_n\}$ of vertices (that is the edges are of the form $x_iy_j$ ) is called tame if it has no $x_iy_jx_ky_l$ path ( $i,j,k,l\in\{ 1,\ldots, n\}$ ) where $j<l$ and $i+j>k+l$ . Calculate the infimum of those real numbers $\alpha$ for which there exists a constant $c=c(\alpha)>0$ such that for all tame graphs $e\le cn^{\alpha}$ , where $e$ is the number of edges and $n$ is half of the number of vertices.

(translated by Miklós Maróti)

Let $A$ be a subgroup of the symmetric group $S_n$ , and $G$ be a normal subgroup of $A$ . Show that if $G$ is transitive, then $|A\colon G|\le 5^{n-1}$ (translated by Miklós Maróti)

Let $A$ be an infinite subset of the set of natural numbers, and denote by $\tau_A(n)$ the number of divisors of $n$ in $A$ . Construct a set $A$ for which $$\sum_{n\le x}\tau_A(n)=x+O(\log\log x)$$ and show that there is no set for which the error term is $o(\log\log x)$ in the above formula.

(translated by Miklós Maróti)

Is it possible to draw circles on the plane so that every line intersects at least one of them but no more than $100$ of them?

Let $f\colon \mathbb{R}^1\rightarrow \mathbb{R}^2$ be a continuous function such that $f(x)=f(x+1)$ for all $x$ , and let $t\in [0,\frac14]$ . Prove that there exists $x\in\mathbb{R}$ such that the vector from $f(x-t)$ to $f(x+t)$ is perpendicular to the vector from $f(x)$ to $f(x+\frac12)$ .

(translated by Miklós Maróti)

Let $S$ be the Sierpiński triangle. What can we say about the Hausdorff dimension of the elevation sets $f^{-1}(y)$ for typical continuous real functions defined on $S$ ? (A property is satisfied for typical continuous real functions on $S$ if the set of functions not having this property is of the first Baire category in the metric space of continuous $S\rightarrow\mathbb{R}$ functions with the supremum norm.)

(translated by Miklós Maróti)

For a given $\alpha >0$ let us consider the regular, non-vanishing $f(z)$ maps on the unit disc $\{ |z|<1 \}$ for which $f(0)=1$ and $\mathrm{Re} z\frac{f'(z)}{f(z)}>-\alpha$ ( $|z|<1$ ). Show that the range of $$g(z)=\frac{1}{(1-z)^{2\alpha}}$$ contains the range of all other such functions. Here we consider that regular branch of $g(z)$ for which $g(0)=1$ .

(translated by Miklós Maróti)

Let $V$ be the set of non-collinear pairs of vectors in $\mathbb{R}^3$ , and $H$ be the set of lines passing through the origin. Is is true that for every continuous map $f\colon V\rightarrow H$ there exists a continuous map $g\colon V\rightarrow \mathbb{R}^3\,\backslash\,\{ 0\}$ such that $g(v)\in f(v)$ for all $v\in V$ ?

(translated by Miklós Maróti)

Let $\zeta_1, \ldots, \zeta_n$ be (not necessarily independent) random variables with normal distribution for which $E\zeta_j=0$ and $E\zeta_j^2\le 1$ for all $1\le j\le n$ . Prove that $$E\left( \max_{1\le j\le n} \zeta_j \right)\le\sqrt{2\log n}$$ (translated by Miklós Maróti)