The Rényi–Ulam pathological liar game with a fixed number of lies

The q-round Rényi–Ulam pathological liar game with k lies on the set [ n ] ≔ { 1 , … , n } is a 2-player perfect information zero sum game. In each round Paul chooses a subset A ⊆ [ n ] and Carole either assigns 1 lie to each element of A or to each element of [ n ] ⧹ A . Paul wins if after q rounds there is at least one element with k or fewer lies. The game is dual to the original Rényi–Ulam liar game for which the winning condition is that at most one element has k or fewer lies. Define F k * ( q ) to be the minimum n such that Paul can win the q-round pathological liar game with k lies and initial set [ n ] . For fixed k we prove that F k * ( q ) is within an absolute constant (depending only on k) of the sphere bound, 2 q / q ⩽ k ; this is already known to hold for the original Rényi–Ulam liar game due to a result of J. Spencer.