Sim the triangle game on K6
Six dots, fifteen lines. Red and Blue take turns colouring one uncoloured line; whoever completes a triangle in their own colour loses. Red moves first. Since R(3,3) = 6 the game can never be drawn, and with perfect play the second player wins — always on the fifteenth move. The computer has solved all 1,350,022 positions; with hints on, each line is marked W or L with the moves left to the end.
the three rules, last blue move
solved
1999 and 2020 claims, re-checked
How the computer knows
A position is two 15-bit masks, the red lines and the blue lines; whose turn it is follows from the counts. Every position reachable without a finished triangle gets one byte in a table indexed in base 3 (315 ≈ 14 million cells): win or loss for the side to move and the moves left with perfect play. A plain depth-first search fills it in about half a second — 1,350,022 positions. The empty board is a loss for Red in 15: Blue can always win, but never before Red is forced to colour the last line.
The three rules
Weronika Wrzos-Kaminska (2020) gave Blue a strategy a person can follow: take the smallest complete subgraph that holds every coloured line and at least one safe line for Blue (the mini-board); (1) only consider Blue's safe lines on it; (2) keep the ones that lie in the most largest blue-safe sets of lines on it; (3) of those, keep the ones in the most largest red-safe sets. Here every Red reply, every mini-board and every tie was followed to the end: the rules never lose. Leave out any one of them and they do.