From d600518989647e1f8183485e6274f1a8afc5ed8f Mon Sep 17 00:00:00 2001
From: Tim Moreton <tim@celo.org>
Date: Thu, 9 Jan 2020 14:42:03 -0800
Subject: [PATCH 1/2] Explain Q and F

---
 consensus/istanbul/validator/default.go | 21 +++++++++++++++++++++
 1 file changed, 21 insertions(+)

diff --git a/consensus/istanbul/validator/default.go b/consensus/istanbul/validator/default.go
index 5ba1849313..8d20fed57a 100644
--- a/consensus/istanbul/validator/default.go
+++ b/consensus/istanbul/validator/default.go
@@ -247,6 +247,27 @@ func (valSet *defaultSet) Copy() istanbul.ValidatorSet {
 	return newValSet
 }
 
+// We include the optimization described at https://arxiv.org/pdf/1901.07160.pdf as “PM-6” and
+// discussed in Lemma 22. For values of N=3F for integer F=1,2,3,.. we can tolerate a quorum
+// size one smaller than anticipated by Q = N - F. The intersection of any two sets of Q
+// nodes of N=3F must contain an honest validator.
+//
+// For example, with N=9, F=2, Q=6. Any two sets of Q=6 from N=9 nodes must overlap
+// by >9-6=3 nodes. At least 3-F=3-2=1 must be honest.
+//
+// 1 2 3 4 5 6 7 8 9
+// x x x x x x
+// 			 y y y y y y
+// 			 F F H
+//
+// For N=10, F=3, Q=7. Any two sets of Q=7 nodes from N=10 must overlap by >4 nodes.
+// At least 4-F=4-3=1 must be honest.
+//
+// 1 2 3 4 5 6 7 8 9 10
+// x x x x x x x
+// 	  	 y y y y y y y
+// 			 F F F H
+
 func (valSet *defaultSet) F() int { return int(math.Ceil(float64(valSet.Size())/3)) - 1 }
 
 func (valSet *defaultSet) MinQuorumSize() int {

From a34eec3f1e6e2d6e106c6c9b8b8ca5884b991c31 Mon Sep 17 00:00:00 2001
From: Tim Moreton <tim@celo.org>
Date: Thu, 9 Jan 2020 14:44:17 -0800
Subject: [PATCH 2/2] Spaces

---
 consensus/istanbul/validator/default.go | 8 ++++----
 1 file changed, 4 insertions(+), 4 deletions(-)

diff --git a/consensus/istanbul/validator/default.go b/consensus/istanbul/validator/default.go
index 8d20fed57a..b4bec35057 100644
--- a/consensus/istanbul/validator/default.go
+++ b/consensus/istanbul/validator/default.go
@@ -257,16 +257,16 @@ func (valSet *defaultSet) Copy() istanbul.ValidatorSet {
 //
 // 1 2 3 4 5 6 7 8 9
 // x x x x x x
-// 			 y y y y y y
-// 			 F F H
+//       y y y y y y
+//       F F H
 //
 // For N=10, F=3, Q=7. Any two sets of Q=7 nodes from N=10 must overlap by >4 nodes.
 // At least 4-F=4-3=1 must be honest.
 //
 // 1 2 3 4 5 6 7 8 9 10
 // x x x x x x x
-// 	  	 y y y y y y y
-// 			 F F F H
+//       y y y y y y y
+//       F F F H
 
 func (valSet *defaultSet) F() int { return int(math.Ceil(float64(valSet.Size())/3)) - 1 }