user134705: Why is the intersection of two quorums must be greater than N/3 when proving Streamlet, a blockchain consensus protocol [BY Chan, Elaine Shi, 2020]?
 
In Streamlet, honest node vote for proposed block in every round. When a block gain vote for more than 2N/3 distinct node, it's notarized. In this context, quorum is the set of distinct votes on a notarized block
Following statement confuses me: The quorum size have to be more than 2N/3 to ensure at most one block can be notarized in a round if the malicious node is less than N/3, because the intersection of two quorums must be greater than N/3 and only malicious block will equivocate.
This intersection greater than N/3 seems come out of nowhere or do I miss something? why can't it be smaller? like N/5 or something else?
This statement and picture are from the material of streamlet in principles of blockchains (ECE 598PV) from illinois and also from the lecture (14:47) given by the author.

user134896: This is just an application of the inclusion exclusion principle. Otherwise, the total number of nodes will be 2n/3+2n/3-(something <1/3) > n.

134705: <p>Why is the intersection of two quorums must be greater than N/3 when proving Streamlet, a blockchain consensus protocol [BY Chan, Elaine Shi, 2020]?</p>
<p><a href="https://i.stack.imgur.com/VHPnf.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/VHPnf.png" alt="The quorum" /></a> <br />
In Streamlet, honest node vote for proposed block in every round. When a block gain vote for more than 2N/3 distinct node, it's notarized. In this context, quorum is the set of distinct votes on a notarized block</p>
<p>Following statement confuses me: The quorum size have to be more than 2N/3 to ensure at most one block can be notarized in a round if the malicious node is less than N/3, because the intersection of two quorums must be greater than N/3 and only malicious block will equivocate.
This intersection greater than N/3 seems come out of nowhere or do I miss something? why can't it be smaller? like N/5 or something else?</p>
<p>This statement and picture are from <a href="https://eprint.iacr.org/2020/088.pdf" rel="nofollow noreferrer">the material of streamlet</a> in principles of blockchains (ECE 598PV) from illinois and also from <a href="https://www.youtube.com/watch?v=8BL1Y-e5VOY&amp;t=920s&amp;ab_channel=indcca" rel="nofollow noreferrer">the lecture</a> (14:47) given by the author.</p>
 134896: <p>This is just an application of the inclusion exclusion principle. Otherwise, the total number of nodes will be 2n/3+2n/3-(something &lt;1/3) &gt; n.</p>
