On the multi-interval Ulam-Rényi game: For 3 lies 4 intervals suffice

Abstract

We study the problem of identifying an initially unknown m-bit number by using yes-no questions when up to a fixed number e of the answers can be erroneous. In the variant we consider here questions are restricted to be the union of up to a fixed number of intervals.For any e ≥ 1 let ke be the minimum k such that for all sufficiently large m, there exists a strategy matching the information theoretic lower bound and only using k-interval questions. It is known that ke = O(e 2 ) and it has been conjectured that the ke = Θ(e). This linearity conjecture is supported by the known results for small values of e as for e ≤ 2 we have ke = e. We focus on the case e = 3 and show k3 ≤ 4 improving upon the previously known bound k3 ≤ 10.