Finite promise games and greedy clique sequences
The finite promise games are a collection of mathematical games developed by American mathematician Harvey Friedman in 2009 which are used to develop a family of fast-growing functions ,  and . The greedy clique sequence is a graph theory concept, also developed by Friedman in 2010, which are used to develop fast-growing functions ,  and . represents the theory of ZFC plus, for each , "there is a strongly -Mahlo cardinal", and  represents the theory of ZFC plus "for each , there is a strongly -Mahlo cardinal". represents the theory of ZFC plus, for each , "there is a -stationary Ramsey cardinal", and  represents the theory of ZFC plus "for each , there is a strongly -stationary Ramsey cardinal". represents the theory of ZFC plus, for each , "there is a -huge cardinal", and  represents the theory of ZFC plus "for each , there is a strongly -huge cardinal". Finite promise games 
Each of the games is finite, predetermined in length, and has two players (Alice and Bob). At each turn, Alice chooses an integer or a number of integers (an offering) and the Bob has to make one of two kinds of promises restricting his future possible moves. In all games, Bob wins if and only if Bob has kept all of his promises. Finite piecewise linear copy/invert games 
Here,  is the set of integers, and  is the set of non-negative integers. Here, all letters represent integers. We say that a map  is piecewise linear if  can be defined by various affine functions with integer coefficients on each of finitely many pieces, where each piece is defined by a finite set of linear inequalities with integer coefficients. For some piecewise linear map , a -inversion of  is some  such that . We then define the game  for nonzero . has  rounds, and alternates between Alice and Bob. At every stage of the game, Alice is required to play , called her offering, which is either of the form  or , where  and  are integers previously played by Bob. Bob is then required to either:

 Accept , thereby playing  and promising that there will be no -inversion of  among the integers ever played by Bob. This promise applies to all past, present and future plays in the game. Reject , thereby play a -inversion of  and promising that  is never played by Bob. In RCA0, it can be proven that Bob always has a winning strategy for any given game. The game  is a modified version where Bob is forced to accept all factorial offers by Alice . Bob always has a winning strategy for  for sufficiently large , although this cannot be proven in any given consistent fragment of , and only . The function  is the smallest  such that Bob can win  for any  such that  and  are greater than or equal to  and all the following values are less than :

 
  (the domain of  is )
 the number of pieces of 
 the absolute values of the coefficients of the inequalities in 
 the absolute values of the coefficients of the affine functions in

Finite polynomial copy/invert games 
Let  be a polynomial with integer coefficients. A special -inversion at  in  consists of  such that . We now define the game  for nonzero , where  are polynomials with integer coefficients. consists of  alternating plays by Alice and Bob. At every stage of the game, Alice is required to play  of the form ,  or , where  is a -tuple of integers previously played by Bob. Bob is then required to either:

 Accept , thereby playing  and promising that there will be no special - or -inversion of  among the integers ever played by Bob. This promise applies to all past, present and future plays in the game. Reject , thereby playing a special - or -inversion of  and promising that  is never played by Bob. Let  be polynomials with integer coefficients. In RCA0, it can be proven that Bob always has a winning strategy for any given game. If  are sufficiently large then Bob wins , which is  where Bob is forced to accept all double factorials  offered by Alice. However, once again, this cannot be proven in any given consistent fragment of , and only . The function  is the smallest  such that Bob can win  for any  such that  and  are greater than or equal to  and all the following values are less than :

 
  (the domain of the polynomials is )
 the degrees of  and 
 the absolute values of the coefficients of  and

Finite linear copy/invert games 
We say that  are additively equivalent if and only if . For nonzero integers  and , we define the game  which consists of  alternating rounds between Alice and Bob. At every stage of the game, Alice is required to play an integer  of the form  or , where  are integers previously played by Bob. Bob is then required to either:

 Accept , thereby playing  and promising that  cannot be written as , where  is additively equivalent to some , and  are integers played by Bob at various times. Reject , thereby playing , where  and  is additively equivalent to some , and promises that  is never played by Bob. Let . In RCA0, it can be proven that Bob always has a winning strategy for any given game. Let . If  is sufficiently large, then Bob wins , where Bob accepts all factorials  offered by Alice. However, once again, this cannot be proven in any given consistent fragment of , and only . The function  is the smallest  such that Bob can win  for any  such that  is greater than or equal to ,  are positive and all the following values are less than :

 
 
 the number of components of each vector in

Functions 
As shown by Friedman, the three functions ,  and  are extremely fast-growing, eventually dominating any functions provably recursive in any consistent fragment of  (one of these is ZFC), but they are computable and provably total in . Greedy clique sequences 
 denotes the set of all tuples of rational numbers. We use subscripts to denote indexes into tuples (starting at 1) and angle brackets to denote concatenation of tuples, e.g. . Given , we define the upper shift of , denoted  to be the result of adding 1 to all its nonnegative components. Given , we say that  and . are called order equivalent if and only if they have the same length and for all ,  iff . A set  is order invariant iff for all order equivalent  and , . Let  be a graph with vertices in . Let  be the set defined as follows: for every edge  in , their concatenation  is in . Then if  is order invariant, we say that  is order invariant. When  is order invariant,  has infinite edges. We are given , , and a simple graph  (or a digraph in the case of upper shift greedy down clique sequences) with vertices in . We define a sequence  as a nonempty tuple  where . This is not a tuple but rather a tuple of tuples. When ,  is said to be an upper shift greedy clique sequence in  if it satisfies the following:

  consists only of zeroes. Let  be an integer such that , or a positive integer if we allow infinite sequences. and let . Then, ,  is not an edge of , and . is a clique in , i.e. contains as an edge every pair of vertices in . When ,  is said to be an upper shift down greedy clique sequence in  if it satisfies the following:

  consists only of zeroes. Let  be an integer such that , or a positive integer if we allow infinite sequences. and let . Then,  or  and  is not an edge of ; and . is a down clique in , i.e. for all  and ,  is an edge of . When ,  is said to be an extreme upper shift down greedy clique sequence in  if it satisfies the following:

  consists only of zeroes. Let  be an integer such that , or a positive integer if we allow infinite sequences. and let . Then,  or  and  is not an edge of ; and . If , then . If , then . If  and , then 
  is a down clique in 

The thread of  is a subsequence  defined inductively like so:

 
 

Given a thread , we say that is open if . Using this Harvey Friedman defined three very powerful functions:

  is the smallest  such that every simple, order invariant graph  has an upper shift greedy clique sequence in  of length at most  with an open thread. is the smallest  such that every order invariant digraph  has an upper shift greedy down clique sequence in  of length at most  with an open thread. is the smallest  such that every order invariant digraph  has an extreme upper shift greedy down clique sequence in  of length at most  with an open thread. and  eventually dominate all functions provably recursive in , but are themselves provably recursive in . eventually dominates all functions provably recursive in , but is itself provably total in . References 

 
 
 

Mathematical games
Mathematical concepts