4. Number theoretic questions

Problem 4.1.
[Snyder] Is there a sense in which a randomly chosen fusion graph doesnâ€™t have cylotomic dimensions? 
Problem 4.2.
[Snyder] Look at all spoke graphs with $N>0$ arms. Are there finitely many $N$tuples $(\ell_1,\dots, \ell_N)$ such that the spoke graph with $N$ arms of lengths $\ell_1,\dots, \ell_N$ has cyclotomic norm squared? 
Problem 4.1.
[Morrison] Is there a positive real number which is a cyclotomic integer and is largest amongst its Galois conjugates, but which is not realized as the dimension of an object in a fusion category?
Remark. [Scott Morrison] The first five such numbers above 2, given in [arxiv:1004.0665], are all known to be realized.

Cite this as: AimPL: Classifying fusion categories, available at http://aimpl.org/fusioncat.