The folder #172 (attempts to) answer this question:

Is it true that in any finite colouring of $\mathbb{N}$ there exist arbitrarily large finite $A$ such that all sums and products of distinct elements in $A$ are the same colour?

The folder #130 aims to answer this question: (Still formalizing this with Aristotle)

Let $A\subset\mathbb{R}^2$ be an infinite set which contains no three points on a line and no four points on a circle. Consider the graph with vertices the points in $A$, where two vertices are joined by an edge if and only if they are an integer distance apart.

How large can the chromatic number and clique number of this graph be? In particular, can the chromatic number be infinite?

The folder #84 aims to (partially) answer this question:

The cycle set of a graph $G$ on $n$ vertices is a set $A\subseteq {3,\ldots,n}$ such that there is a cycle in $G$ of length $\ell$ if and only if $\ell \in A$. Let $f(n)$ count the number of possible such $A$.

Prove that $f(n)=o(2^n)$.

Prove that $f(n)/2^{n/2}\to \infty$.

#81 (Still a partial result here)

Let $G$ be a chordal graph on $n$ vertices - that is, $G$ has no induced cycles of length greater than $3$. Can the edges of $G$ be partitioned into $n^2/6+O(n)$ many cliques?

Problem(s) to try sometime:

197 (possibly solved pending feedback - still needs a damn lot of fixing!)

111

538

68

455

112