We will survey the class of linear groups, and identify three methods of constructing linear groups: algebraic, geometric, and arithmetic. We will propose that every subgroup comes from one of these types of constructions. There are many interesting consequences all across group theory and geometry if this is indeed the case. Focusing on the case of 2-by-2 matrix groups, the algebraic and arithmetic subgroups are very well-understood. We will fill in some understanding of geometric subgroups in this setting, by showing that for any prime p, there is a closed surface group in \({\rm PSL}_2(\mathbb{Z}[1/p])\). These surface groups will have genus proportional to \(p\), and we speculate that the construction is optimal.

Classical extremal results in graph theory (such as Turán's theorem) concern the maximal size of of a graph of given order and without certain subgraphs. Bollobás, Erdős, and Szemerédi in 1975 studied extremal problems in multipartite graphs. One of their problems (in its complementary form) was determining the maximal degree of a multipartite graph without an independent transversal. This problem has received considerable attention and was settle in 2006 (Szabó--Tardos and Haxell--Szabó). Other questions asked by Bollobás, Erdős, and Szemerédi remain open, such as determining:

(1) the maximum degree in a multipartite graph without a partial independent transversal, and;
(2) the minimum degree that forces an octahedral graph in balanced tripartite graphs.

In this talk I will survey recent progress on these and other related problems.