Formula omitted]-ary Rényi–Ulam pathological liar game with one lie

Abstract

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