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Let $p>q$ be positive integers, set $P=\{1,2,...,p\}$, and $Q\subset P$ be an unknown set of $q$ elements.

In every step, we can guess a set $T\subset P$ and then know the size $|T\cap Q|$ of its intersection with $Q$. What is the minimal number of steps, $S(p,q)$, in which we are guaranteed to find $Q$?

I want to prove the following lemma.

Lemma. $P_1,P_2$ are sets of $p_1,p_2$ elements disjoint, given $|P_i\cap Q|=r_i,i=1,2$, note the minimal step number to surely get Q be $S(p_1,p_2;r_1,r_2)$, then $S(p_1,p_2;r_1,r_2)=S(p_1,r_1)+S(p_2,r_2)$.

LLLL
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    Should $|T \cap Q|$ increase? Why is not possible to choose $T = Q$ at the first step? – dcolazin Dec 13 '24 at 16:26
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    @dcolazin You do not know what $Q$ is. You are only told what $|T\cap Q|$ is. – Mike Earnest Dec 13 '24 at 16:56
  • Do $P_1,P_2$ partition $P$? If so, it seems kinda obvious: you use an algorithm $A_1$ that attains $S(p_1,r_1)$ to figure out the part of $Q$ that intersect $P_1$, then analogously $A_2$ and $P_2$. – Benjamin Wang Dec 13 '24 at 17:08
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    Are you given $q$ at the start? You can always get it in one more guess by taking $T=P$ so that doesn't make much difference. – Ross Millikan Dec 13 '24 at 17:15
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    If I understand correctly, we are given $p$ and $q$, and we must find $Q$ in the minimal amount of steps. So $S(p,0) = 0$, $S(p,p) =0$, and $S(p,1) = S(p,p-1)= \lceil \log_2(p)\rceil$ (or maybe this $-1$) by dichotomic search. Is that right ? Or is guessing $q$ part of the game ? Then how exactly is $S$ defined : what exactly should be minimized : the worst-case performance ? The average performance ? – Martin Dec 13 '24 at 17:56
  • If $q$ were unknown, you would be in the situation of guessing the answers to a $p$-question true or false test, given only your score after each attempt. This takes around $2p/\log p$ steps, as shown in this question. – Mike Earnest Dec 13 '24 at 18:36
  • @Benjamin Wang This only shows $S(p1,p2;r1,r2)\leq S(p1,r1)+S(p2,r2)$. – LLLL Dec 14 '24 at 04:18
  • @Martin Yes you are right! What I mean is for each strategy R, there exists a worst case maximize the step number, note it S(R), and I want to find the best strategy to minimize S(R). – LLLL Dec 14 '24 at 04:24

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