Littlewood asked for the maximum number N of congruent infinite cylinders that can be arranged in R^3 so that every pair touches. In this talk, we discuss the techniques behind some recent improvements on the upper bounds for this problem, culminating in our recent result showing that N \leq 10. Paired with a lower bound by Bozóki, Lee, and Rónyai, this establishes that N is in \{7,8,9,10\}. Our method is based on linear algebra and Ramsey theory, and makes partial use of computer verification. Based on joint work with Travis Dillon and Junnosuke Koizumi.