Rounds in a combinatorial search problem
  We consider the following combinatorial search problem: we are given some excellent elements of $[n]$ and we should find at least one, asking questions of the following type: "Is there an excellent element in $A \subset [n]$?". G.O.H. Katona proved sharp results for the number of questions needed to ask in the adaptive, non-adaptive and two-round versions of this problem.   We verify a conjecture of Katona by proving that in the $r$-round version we need to ask $rn^{1/r}+O(1)$ queries for fixed $r$ and this is sharp.   We also prove bounds for the queries needed to ask if we want to find at least $d$ excellent elements. 
D\'aniel Gerbner, M\'at\'e Vizer
math.CO cs.DS