A fork (title of the episode) is a tactic in chess in which a piece attacks two or more pieces simultaneously. In the first speed chess game Beth played against Benny Watts, Watts managed to fork Beth with his knight, attacking her queen and king simultaneously. Because this also gives a check, Beth is forced to move her king, leading to Watts taking her queen with his knight. Beth resigned after this.
During the meeting in the students' club, Benny Watts introduces Beth to two young players named Danny Weiss and David Friedman. This might be a nod to Dan Weiss and David Benioff (whose birth name is David Friedman), co-creators of the HBO fantasy epic Game of Thrones. Scott Frank, the creator of this show, was mentioned by Benioff and Weiss as their friend several times when they announced their feature film "Dirty White Boys," claiming Frank was the one who recommended the book to them. Incidentally, Thomas Brodie-Sangster, who plays Benny Watts, appeared on Game of Thrones as Jojen Reed.
The exterior scenes of the Ohio university campus were in fact shot in Aurora, Ontario, Canada on the campus of St. Andrew's College, an all-boy private high school. Fun fact, old boys of this school are actors Kiefer Sutherland (24, Designated Survivor) and Stephen Amell (Arrow, Heels).
When Beth and Beltik are training he goes to the bathroom. There he speaks to himself about being stupid trying to hit Beth and finds the Librium. When he goes out, she's dancing. A nice homage to Pulp Fiction when Vincent is with Mia but without the lethal overdose.