On the R\'enyi-Ulam Game with Restricted Size Queries

TL;DR

This paper studies a variant of the Re9nyi-Ulam game where queries are restricted in size and the Responder can lie a limited number of times, providing bounds and exact values for the minimum queries needed.

Contribution

It introduces the Convexity Lemma, offers bounds for the game, and generalizes previous results to cases with multiple lies.

Findings

Established a general lower bound for $RU_e9 lap{e9}l^k(n)$

Provided an upper bound close to the lower bound, differing by at most $2e7+1$

Derived exact values for large $n$ relative to $k$

Abstract

We investigate the following version of the well-known R\'enyi-Ulam game. Two players - the Questioner and the Responder - play against each other. The Responder thinks of a number from the set $\{1,\ldots,n\}$, and the Questioner has to find this number. To do this, he can ask whether a chosen set of at most $k$ elements contains the thought number. The Responder answers with YES or NO immediately, but during the game, he may lie at most $\ell$ times. The minimum number of queries needed for the Questioner to surely find the unknown element is denoted by $RU_\ell^k(n)$. First, we develop a highly effective tool that we call Convexity Lemma. By using this lemma, we give a general lower bound of $RU_\ell^k(n)$ and an upper bound which differs from the lower one by at most $2\ell+1$. We also give its exact value when $n$ is sufficiently large compared to $k$. With these, we managed to improve and generalize the results obtained by Meng, Lin, and Yang in a 2013 paper about the case $\ell=1$.