%------------------------------------------------------------------------------
% File     : GRA121+1 : TPTP v9.3.1. Released v9.3.0.
% Domain   : Graph Theory
% Problem  : Ramsey number r(3,4) = 9 with Peano axioms
% Version  : Especial.
% English  : The Ramsey property that if an undirected graph has at least
%            R(m,n) vertices, then it must have either an m-clique or an
%            n-anticlique.

% Refs     : [Ram30] Ramsey (1930), On a Problem of Formal Logic
%          : [Kal24] Kalizyk (2024), Email to G. Sutcliffe
% Source   : [Kal24]
% Names    : r_3_4_9_p [Kal24]

% Status   : Unsatisfiable
% Rating   : 0.86 v9.3.0
% Syntax   : Number of formulae    :    5 (   1 unt;   0 def)
%            Number of atoms       :   23 (  12 equ)
%            Maximal formula atoms :   12 (   4 avg)
%            Number of connectives :   24 (   6   ~;  18   |;   0   &)
%                                         (   0 <=>;   0  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   16 (   8 avg)
%            Maximal term depth    :    2 (   1 avg)
%            Number of predicates  :    2 (   1 usr;   0 prp; 2-2 aty)
%            Number of functors    :    2 (   2 usr;   1 con; 0-1 aty)
%            Number of variables   :   12 (  12   !;   0   ?)
% SPC      : FOF_UNS_RFO_SEQ

% Comments :
%------------------------------------------------------------------------------
fof(rsym,axiom,
    ! [X,Y] :
      ( ~ r(X,Y)
      | r(Y,X) ) ).

fof(rno3cl,axiom,
    ! [X,Y,Z] :
      ( X = Y
      | X = Z
      | Y = Z
      | ~ r(X,Y)
      | ~ r(X,Z)
      | ~ r(Y,Z) ) ).

fof(rno4acl,axiom,
    ! [X,Y,Z,W] :
      ( X = Y
      | X = Z
      | Y = Z
      | X = W
      | Y = W
      | Z = W
      | r(X,Y)
      | r(X,Z)
      | r(Y,Z)
      | r(X,W)
      | r(Y,W)
      | r(Z,W) ) ).

fof(pfnc,axiom,
    ! [X] : f(X) != c ).

fof(pfinj,axiom,
    ! [X,Y] :
      ( f(X) != f(Y)
      | X = Y ) ).

%------------------------------------------------------------------------------