We investigate the following version of the well-known Rényi–Ulam game. Two players – the Questioner and the Responder – play against each other. The Responder thinks of a number from the set $\{1,\dots ,n\}$, 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 $R{U}_{\ell }^k\left(n\right)$. First, we develop a highly effective tool that we call Convexity Lemma. By using this lemma, we give a general lower bound of $R{U}_{\ell }^k\left(n\right)$ 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 $\ell =1$.

我们调查了著名的 Rényi–Ulam 博弈的以下版本。两名玩家——提问者和回应者——互相对抗。响应者从集合中想到一个数字$\{\mathrm{1个},\dots \dots ,n\}$，发问者必须找到这个数字。要做到这一点，他可以询问是否选择了一组至多$k$元素包含思想编号。Responder 立即回答 YES 或 NO，但在游戏过程中，他最多可以说谎$\ell$次。发问者确定找到未知元素所需的最少查询次数表示为$R{ü}_{\ell }^k\left(n\right)$. 首先，我们开发了一个高效的工具，我们称之为凸性引理。通过使用这个引理，我们给出了一个一般的下界$R{ü}_{\ell }^k\left(n\right)$以及最多与下限不同的上限$\mathrm{2个}\ell +\mathrm{1个}$. 我们也给出了它的确切值$n$与相比足够大$k$. 有了这些，我们设法改进和概括了 Meng、Lin 和 Yang 在 2013 年关于该案例的论文中获得的结果$\ell =\mathrm{1个}$.