Washington, 20, averaged 15.2 points and 7.6 rebounds in 35 games (33 starts) during his second season with the Wildcats.
Gilgeous-Alexander averaged 14.4 points and 5.1 assists per game on a Wildcats team that registered a 26-11 record last season.
New York holds the No. 9 overall pick after finishing last season with a 29-53 record.
Kentucky will have PJ Washington for another season, but Wenyen Gabriel has opted to remain in the NBA Draft.
Travis averaged 19.5 points and 8.7 rebounds per game for a Stanford team that went 19-16 last season.
Walker, a native of Lake Wales, Fla., made five tackles in 12 games as a sophomore in 2017.
Killeya-Jones is a former 5-star recruit who played a limited role for Kentucky this past season.
Tennessee and Alabama were the richest of the rich, topping the $100 million mark in gross revenues for the fiscal year 2016-17.
Tyrese Maxey, a top-15 guard in the class of 2019, has selected Kentucky.