On the Multi-Interval Ulam-R\'enyi Game: for 3 lies 4 intervals suffice

TL;DR

This paper investigates the minimal number of interval questions needed to identify an unknown number with up to three erroneous answers, improving bounds from 10 to 4 intervals for the case of three errors.

Contribution

It extends previous results by proving that four intervals suffice for three errors, supporting the linearity conjecture in the multi-interval Ulam-Rényi game.

Findings

Established that 4 intervals suffice for 3 errors.

Improved the upper bound from 10 to 4 intervals.

Supports the linearity conjecture for the number of intervals needed.

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 $e \geq 1$ let $k_e$ 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 $k_e = O(e^2)$. However, it has been conjectured that the $k_e = \Theta(e).$ This linearity conjecture is supported by the known results for small values of $e$. For $e\leq2$ we have $k_e = e.$ We extend these results to the case $e=3$. We show $k_3 \leq 4$ improving upon the previously known bound $k_3 \leq 10.$