 So, if the event A is common knowledge in state of the world omega star, suppose the event A is common knowledge in the state of the world omega star and another you have another state omega dash which is in the same element of the partition as omega star for some player i. So, an omega dash belongs to F i of omega star for some player i. Then what can we say about this? Then event A is also common knowledge, it is also common knowledge in omega dash. Now, this may seem very strong and very surprising essentially what this means is. So, what is this theorem claiming? If event A is common knowledge in the state of the world omega star in some state of the world omega star and there is some player who cannot distinguish between omega star and omega dash. Then even if omega dash has occurred then even in that state omega dash the event A has to be common knowledge. This seems super strong because it seems like then you know how is this because one fellow cannot distinguish between omega dash and omega star. How can this country such a thing hold? How can it be that A remains common knowledge? But that is exactly how the criterion of common knowledge is. It is extremely demanding you know the very that it must be that player i knows that player d knows that etcetera for every such sequence of players. That criterion effectively implies this particular thing. So, if there is any one player who cannot distinguish between two events then the entire thing for it to be common entire for the event to be common knowledge then it has to be that it is also common knowledge in that other state that this player cannot distinguish between. So, let us just look at the proof. So, A is common knowledge in omega star. So, now consider any sequence of like this i 0 I will start from i 0 i 1 dot dot dot i k and then it has to be that omega star belongs to k i 0 of k i 1 dot dot dot let us call this R k i R. This is because A is common knowledge in omega star. So, this is true for every i 0 to i i R. Now, omega star belongs to this. Now what I will do is I will take i 0 to be that player i. The player i who cannot distinguish between omega star and omega dash. So, take i 0 is equal to i that means take it to be this guy. Now this if omega star belongs to k i 0 of all this then it means therefore that f i of omega star is a subset of k i 1 of k i 2 of dot dot dot k i R of A. But then this belongs to a subset this is a subset of all this which means what which means omega dash is an element of k i 1 of k i 2 dot dot dot k i R of A. And remember I just took an arbitrary sequence like this i 0 i 1 dot dot dot i R now and I took i 0 as i but the rest of them I can vary in whatever way I want. So, I effectively means that this is true therefore for this whole thing is true for holds for all i 1 to i R. The point is that because I am not allowed to take any length of sequence all I am doing is I am fixing the first player as my player i and then the remaining I am just varying over the remaining and then I will get any in any length of sequences from there also. So, this is true for all i 1 which means that A is common knowledge in omega dash. So, this also means that you know as another consequence of the demanding nature of common the definition of common knowledge is that you also have this particular property that if if A is common knowledge in omega star and A is contained in B. So, A is more specific B is more general if A is contained in B then what does this mean if A is common knowledge in omega star and A is contained in B then B would also B is also common knowledge in omega star. So, now let us discuss the structure of sets that turn out to be common knowledge and there is a very. So, if you see this properties now you realize that you know that for a set for an event to be common knowledge it has to have some very specific structure because it is you know it seems like that is there not every set can end up being common knowledge. So, what so for the first observation towards that is first is we can go back to we can see what what he said and let us look at actually what is the if you are you have a certain partition like this you have a partition like this and suppose I gave you a set I gave you a set like this. So, this is my set A the red boundary is my set A now what is Ki of A if this if this is the partition of player I what would be Ki of A it is just this likewise and then suppose if I suppose I give you another partition like this I gave you another partition like this now can you tell me what is for this player what would be is knowledge of A those two. So, I will highlight that here let me see if this color works it would be this. So, what this means is that Ki of A is always the union of elements of a partition for every player. Now, when you want a set to be when you want an event to be common knowledge for that event to be common knowledge it has to be no firstly it has to be knowledge it has to be known to every player and when is it known to every player when it contains in it the entire element of a particular an entire element or one or more entire elements of a partition of the partition of that player. So, what this means is for a set to end up becoming common knowledge it should be structured very nicely with the way the partitions are already defined otherwise you will soon into a sufficiently high into the hierarchy you will have a problem that you know someone knows someone does not know that someone knows that someone knows something. So, that is what we will come to now. So, let us try to write out this condition. So, characterizing that are common knowledge. Now, what we will do is we will will define a graph define a graph this define a graph G and the word this is defined on vertices which are the states of the world. Take the vertices as the states of the world and when do we define when is there an edge between two vertices if there is an edge between two vertices if there is at least one there is some player who cannot distinguish between them. So, there is an edge between omega, omega dash in y if there exists player i in n such that omega dash belongs to F i of omega. So, player if there is at least one player who cannot tell the difference between omega and omega dash. So, this condition remember is symmetric omega dash belongs to F i of omega is the same as saying omega belongs to F i of omega dash. This is these are equivalent. So, this basically gives you a graph. So, this will give you a graph in which in which there are these vertices which is the states of the world you join any you join a pair of these vertices if there is at least one player who cannot tell the difference between the two. So, this this pillar cannot tell the difference between these two this fellow cannot tell the difference between these two there will be a third player who cannot tell the difference between these etcetera. So, the subset C of y is said to be is called a connected component is called a connected component if for all omega, omega dash and C there exists a path connecting omega to omega dash in the graph G ok. So, you take your set of vertices, you join them in this way that whenever there is some fellow who cannot distinguish between them you draw an edge ok. Now, it can happen that the entire graph becomes connected that means between you get a path from any vertex to any other vertex. Path means what? It is just a collection of edges one starting from you start from i to j go from j to k from k to l etcetera and eventually end up at the vertex that you wanted to. So, starting so if that it could happen that there is a path between and every pair of vertices or it can happen that the graph has multiple connected components means that there is a subset of vertices such that they all share have a path between each of them, but then none of them have a path with another subset ok. So, the so a connected component will always be like this. So, you could have a graph like this where there is one connected component of this kind. So, all of these guys there is a they are related in this way that there is a path between them, but none of them have an edge to something in this subset because if there was an edge like this from here to here right then this would become one connected component actually this would not be two separate connected components this is clear. So, that is because then there would obviously be a path from any one vertex from here to here. So, connected component is one such that for if you so you can of course you can make so usually when we talk of connected component. So, I will come to maximality and so on see the point here is that between every two vertices there has to be a path. Now, you can of course take a very small connected component which is the point he is making. So, I can take so for example these guys and say well this is a connected one yes it is a connected component, but can it be enlarged in further yes it can be you can enlarge it to the point where you all of these guys are included, but you cannot enlarge it further than this. So, actually formally speaking I connected component is one where there is a path between any two vertices and if you take a vertex and you cannot add another vertex to this and still keep it a connected component. So, this is a connected component now and we can talk of a maximal connected component or let us maybe let us just extend this definition itself. So, C is called a connect so I have just extended the earlier definition because I did not want to create another maximal connect this thing. So, let C be a connect as set C is a connected component if there is a path between every any two vertices and there is no edge connecting a vertex in C to a vertex outside C. Then such a set is called a connected component. So, the graph that we have created here could end up having one or more connected component. Now, if the entire graph is connected then it has one connected component and that is the graph itself other, but in general it could have multiple connected component because every, otherwise every pair of vertices is a connected component, trivially a connected component. So, the theorem is this. So, the structure of sets that end up being common knowledge. So, let us first actually before the theorem let me define this, define C of omega as the connected component containing omega. So, the entire graph breaks down into connected components and every vertex is going to be in one and exactly one such connected component. Now, let C of omega be that connected component for a vertex omega. Now, the theorem is this that an event A is common knowledge in omega star let us say if and only if A is a subset of C of omega star sorry super set of super set of C of the connected component is a subset of the event. A is a super set of C of omega star. Now, what this means is that your A should be general enough that so how did we create the graph remember we created the graph by saying if there is any one player who cannot distinguish between two events then there is an edge between them. Now, you keep creating this chain of players you know this fellow cannot distinguish between this two that they cannot distinguish between that and something else etc. And you create a graph in that you will find a connected component. So, for A to be a common knowledge it has to be large enough or more you know but intuitively general enough that the entire connected component for the event that you are considering for the state of the world that you are considering that entire connected component has to be in A. So, what this means is for us for when a set is ends up being common knowledge it cannot be very, very specific. This is also intuitively correct you know the things that everybody knows that everybody knows that everybody knows etc. are usually things that are generalities you know that today is Friday you know that such now is this time etc. If you ask me something very specific that means that is known to very specific players who have that specific knowledge who have the information channels to know that particular specific thing or something to be known by common knowledge across multiple players it has to be where for large enough or in general basically general enough and a trivial set therefore that is always common knowledge is why itself right everybody knows that this is this is this is all the states of the world or in short everybody knows that something has happened is it clear. So, next time we will prove this another consequence of this is that in any state of the world there is always one set that is there is one event that is always common knowledge and the minimal such event that is common knowledge the minimal thing that everybody knows that everybody knows that everybody knows is that connected component is that is the most specific thing that everyone can have agreement on. So, this is the notion of common knowledge has tremendous implications like for example take all these coordination problems consensus you know coordination or distributed optimization federated learning etcetera what do various agents in a actually know about a particular thing is being is basically being expressed through this you can actually show that there are certain types of protocols that cannot converge unless there is an underlying structure where in a certain thing is common knowledge you know if for example if if if there is if if certain elements are not common knowledge like for example players do not agree on on certain distributions or to not agree on certain aspects of the data or something like that then certain there are you can show that there is no protocol that can work very interesting things can be shown shown from this.