The dichromatic number of a tournament is the minimum number of colors needed to color the vertices so that no directed cycle is monochromatic. Nguyen, Scott, and Seymour (2025) asked whether every tournament with dichromatic number 2k has a vertex whose out-neighborhood induces a subtournament of dichromatic number at least k. We disprove this, by finding a counterexample with dichromatic number Omega(k^2). Furthermore, this quadratic bound is tight up to a constant factor; we show that for every integer k > 0, every tournament of dichromatic number Omega(k^2) contains a vertex whose out-neighborhood induces a subtournament of dichromatic number more than k. This improves the previous best bound O(k^2 log k) due to Colinot and Newman (2026). To show this, we prove that every tournament of fractional dichromatic number c has a dominating set of size at most 5c-1. The proof uses Brouwer's fixed point theorem. This is joint work with Seokbeom Kim.