QQ-ary Rényi–Ulam pathological liar game with one lie

The mm-round qq-ary Rényi–Ulam pathological liar game with  eelies  , referred to as the game [q,e,n;m]∗[q,e,n;m]∗, is considered. Two players, say Paul and Carole, fix nonnegative integers mm, nn, qq and ee. In each round, Paul splits [n]≔{1,2,…,n}[n]≔{1,2,…,n} into qq subsets, and Carole chooses one subset as her answer and assigns 11 lie to all elements except those in her answer. Paul wins  , after mm rounds, if there exists at least one element assigned with ee or fewer lies. Let f∗(q,e,n)f∗(q,e,n) be the maximum value of mm such that Paul can certainly win the game [q,e,n;m]∗[q,e,n;m]∗. This paper gives the exact value of f∗(q,1,n)f∗(q,1,n) for n≥qq−1n≥qq−1 and presents a tight bound on f∗(q,1,n)f∗(q,1,n) for n