A Linear Lower Bound for Dominating Sets in $k$-Majority Tournaments

A $k$-majority tournament on a finite vertex set is defined by $2k-1$ linear orders, with $u\to v$ when $u$ lies above $v$ in at least $k$ of the orders. Let $F(k)$ be the maximum, over all $k$-majority tournaments, of the size of a minimum dominating set. Alon, Brightwell, Kierstead, Kostochka, and Winkler proved that $C_1k/\log k \leq F(k) \leq C_2k\log k$ for suitable positive constants $C_1$ and $C_2$. In this paper, we prove the linear lower bound $F(k)\ge \left\lfloor\frac{k+1}{2}\right\rfloor$ for $k\ge 3$.