  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 $\ell=1$.
