# Hypercube graph

Let $Q_n$ be the hypercube graph.

• $Q_1 = K_2$ (the complete graph with two vertices),
• $Q_2 = C_4$ (the square), and
• $Q_n$ is constructed taking two copies of $Q_{n-1}$ and adding an edge that connects each vertex of the first copy with the corresponding vertex of the second copy.

Easy: Find $\chi(Q_n)$ (chromatic number) for every $n$.

Harder: Find every $n$ such that $Q_n$ is planar.

# A generic curve

Exhibit a curve in $\Bbb{R}^n$ such that any $(n+1)$-uple of points in its trace is in general position.

# An indecomposable module

Suppose $A=k[x_1,\dots,x_n]$ and let $I$ be a proper ideal containing all monomials of degree $\geq N$. Then $A/I$ is an indecomposable $A$-module.

# A generalization of Schur’s lemma

(Dixmier) Let $A$ be a $\Bbb{C}$-algebra (in fact, any uncountable algebraically closed field will do) and $V$ be a simple $A$-module having a countable basis as a $\Bbb{C}$-vector space. Then $D=\mathrm{End}_A(V)\simeq \Bbb{C}$.

Brief outline:

1. By Schur’s lemma, $D$ is a division algebra. Moreover, it has a countable basis.
2. Suppose $\varphi\in D$ does not act as a scalar. Then $\varphi$ is trascendental over $\Bbb{C}$.
3. This is absurd since it would imply $\Bbb{C}(\varphi)\subseteq D$ has uncountable dimension.

# One too many

Consider a permutation $\sigma$ of $mn+1$ letters written in one line notation, say $\sigma = a_1\ldots a_{mn+1}$. An increasing subsequence of $\sigma$ is a choice of integers $i_1 <\cdots < i_k$ such that $a_{i_1}< \cdots < a_{i_k}$, a decreasing subsequence is defined analogously. Show that any permutation in $mn+1$ letters has either an increasing subsequence of length $n+1$ or a decreasing subsequence of length $m+1$. Is the bound optimal?

# Too large to measure

Let $X$ be an infinite dimensional normed space. There is no non-trivial translation invariant Borel measure on $X$ which is finite on open balls.

# Spooky geometry IV

Let $[a,b]$ and $[c,d]$ be subintervals of $[0,1]$ and call them non-overlapping if they intersect only possibly in their endpoints. Construct a continuous curve $\sigma:[0,1]\to H$ with $H$ a Hilbert space, such that if $[a,b]$ and $[c,d]$ are any non-overlapping subintervals of $[0,1]$, then $\sigma(b)-\sigma(a)$ and $\sigma(d)-\sigma(c)$ are orthogonal.

Harder: show that this can be carried out in any infinite-dimensional Hilbert space.