gap('LoadPackage("QPA");')

import tempfile as _tf, os as _os
_gap_code = r"""
DeclareOperation("NthRadical",[IsList]);
InstallMethod(NthRadical, "for a representation of a quiver", [IsList], 0, function(LIST)
    local M, N, f, h, i, n;
    M := LIST[1];
    n := LIST[2];
    if n = 0 then
        return(IdentityMapping(M));
    else
        f := RadicalOfModuleInclusion(M);
        N := Source(f);
        for i in [1..n-1] do
            h := RadicalOfModuleInclusion(N);
            N := Source(h);
            f := \*(h,f);
        od;
        return(f);
    fi;
end);

DeclareOperation("ARQuiverNak",[IsList]);
InstallMethod(ARQuiverNak, "for a representation of a quiver", [IsList], 0, function(LIST)
    local A, UU, i, injA, j;
    A := LIST[1];
    injA := IndecInjectiveModules(A);
    UU := [];
    for i in injA do
        for j in [0..Dimension(i)-1] do
            Append(UU,[Source(NthRadical([i,j]))]);
        od;
    od;
    return(UU);
end);

DeclareOperation("numbersimplespdatleast3",[IsList]);

InstallMethod(numbersimplespdatleast3, "for a representation of a quiver", [IsList],0,function(LIST)
    local A, UU, simA;
    A := LIST[1];
    simA := SimpleModules(A);
    UU := Filtered(simA,x->ProjDimensionOfModule(x,30)>=3);
    return(Size(UU));
end);

DeclareOperation("IsNtorsionfree",[IsList]);

InstallMethod(IsNtorsionfree, "for a representation of a quiver", [IsList],0,function(LIST)
    local A, CoRegA, M, i, n, temm23;
    A := LIST[1];
    M := LIST[2];

    n := LIST[3];

    CoRegA := DirectSumOfQPAModules(IndecInjectiveModules(A));
    temm23 := [];
    for i in [0..n-1] do Append(temm23,[Size(ExtOverAlgebra(NthSyzygy(CoRegA,i),DTr(M))[2])]);
    od;
    return(Sum(temm23));
end);

DeclareOperation("topreflexive",[IsList]);

InstallMethod(topreflexive, "for a representation of a quiver", [IsList],0,function(LIST)
    local A, LL, LL2, UU, UU2, x;
    A := LIST[1];
    LL := ARQuiverNak([A]);
    LL2 := Filtered(LL,x->IsProjectiveModule(x)=false and IsNtorsionfree([A,x,2])=0);
    UU := [];
    for x in LL2 do Append(UU,[DimensionVector(TopOfModule(x))]);
    od;
    UU2 := Set(UU);
    return(Size(UU2));
end);
"""
with _tf.NamedTemporaryFile(mode="w", suffix=".g", delete=False, dir="/tmp") as _f:
    _f.write('LoadPackage("QPA");;\n')
    _f.write(_gap_code)
    _tmp = _f.name
gap.eval('Read("' + _tmp + '");')
_os.unlink(_tmp)

def kupisch(D):
    DR = D.reverse()
    H = DR.heights()
    return [1 + H[i] for i, s in enumerate(DR) if s == 0] + [1]

def statistic(D):
    K = kupisch(D)
    A = gap.NakayamaAlgebra(gap.GF(3), K)
    return ZZ(gap.topreflexive([A]))

N_vertices = 13 # the maximal number of vertices we consider
N_Ramsey = 7 # all graphs with Ramsey number at most N_Ramsey
statistic_dict = dict()
def statistic(G):
    return statistic_dict.get(G.canonical_label().copy(immutable=True))

"""
The Ramsey number of a graph.

This is the smallest integer $n$ such that every two-colouring of the
$n$ vertices of the complete graph $K_n$ contains a (not necessarily
induced) monochromatic copy of the given graph. [1]

Thus, the Ramsey number of the complete graph $K_n$ is the ordinary Ramsey number $R(n,n)$.  Very few of these numbers are known. [2,3]

[1] Chvatál,Rödl,Szemerédi,Trotter, The Ramsey number of a graph with bounded maximum degree. doi:10.1016/0095-8956(83)90037-0

[2] Radziszowski, Stanisław P. "Small Ramsey numbers." Electron. J. Combin 1.7 (1994).

[3] [[wikipedia:Ramsey's theorem#Ramsey numbers]]

[4] Hendry, G. R. T. Ramsey numbers for graphs with five vertices

"""
def check_Ramsey(n, already_found=[]):
    r"""
    Colour the complete graph on n vertices with two colours, and
    return the subgraphs which appear in all colourings.

    EXAMPLES::

        sage: L = dict()
        sage: n = 2; L[n] = check_Ramsey(n)
        sage: n = 3; L[n] = check_Ramsey(n, sum(L.values(), []))
        sage: n = 4; L[n] = check_Ramsey(n, sum(L.values(), []))
        sage: n = 5; L[n] = check_Ramsey(n, sum(L.values(), []))
        sage: n = 6; L[n] = check_Ramsey(n, sum(L.values(), []))
        sage: n = 7; L[n] = check_Ramsey(n, sum(L.values(), []))

        sage: n=6; graphics_array([H.plot() for H in L[n] if H.is_connected()])
    """
    def has_copy(H1, H2, H):
        r"""
        Return True if there is a copy of H in H1=s or H2=E\s.
        """
        S1 = H1.subgraph_search(H)
        if not S1 is None:
            return True
        S2 = H2.subgraph_search(H)
        if not S2 is None:
            return True
        return False

    candidates = [H.canonical_label() for k in range(n+1) for H in graphs(k)]
    candidates = [H for H in candidates if H not in already_found]
    G = graphs.CompleteGraph(n)
    E = Set(G.edges(labels=False))
    V = G.vertices()
    for size in range(1+len(E)//2):
        print("check subgraphs of size", size)
        tested = []
        for s in E.subsets(size):
            H1 = Graph(n)
            H1.add_edges(s)
            H1 = H1.canonical_label()
            H2 = Graph(n)
            H2.add_edges(E.difference(s))
            H2 = H2.canonical_label()
            if (H1, H2) not in tested:
                tested += [(H1, H2)]
                candidates = [H for H in candidates if has_copy(H1, H2, H)]

    return candidates

def add_to_dict(G, v):
    H = G.canonical_label().copy(immutable=True)
    if H in statistic_dict:
        assert statistic_dict[H] == v, "The graph %s should have Ramsey number %s, got %s instead"%(repr(H), statistic_dict[H], v)
        print("%s already known to have Ramsey value %s"%(repr(H), v))
    else:
        statistic_dict[H] = v

Ramsey_small = dict()
for n in range(N_Ramsey+1):
    Ramsey_small[n] = check_Ramsey(n, sum(Ramsey_small.values(), []))

for v, lG in Ramsey_small.items():
    for G in lG:
        add_to_dict(G, v)

# table IIIa
Ramsey_almost_complete = dict()
Ramsey_almost_complete[3] = 3
Ramsey_almost_complete[4] = 10
Ramsey_almost_complete[5] = 22

for n, v in Ramsey_almost_complete.items():
    G = graphs.CompleteGraph(n)
    G.delete_edge(G.edges()[0])
    add_to_dict(G, v)

# G8 - G20 are from Table 1 in
# Hendry, G. R. T. Ramsey numbers for graphs with five vertices

G8 = Graph([(1,2),(2,3),(3,4),(4,5),(3,5)]).canonical_label().copy(immutable=True)
G9 = Graph([(1,2),(2,3),(3,4),(4,5),(2,4)]).canonical_label().copy(immutable=True)
G10 = Graph([(1,2),(1,3),(1,4),(1,5),(2,3)]).canonical_label().copy(immutable=True)
G12 = Graph([(1,2),(2,3),(3,4),(4,5),(1,3),(3,5)]).canonical_label().copy(immutable=True)
G13 = Graph([(1,2),(1,3),(1,4),(1,5),(2,3),(3,4)]).canonical_label().copy(immutable=True)
G14 = Graph([(1,2),(1,3),(1,4),(2,5),(2,3),(3,4)]).canonical_label().copy(immutable=True)
G15 = Graph([(1,2),(2,3),(3,4),(4,5),(5,1),(1,3)]).canonical_label().copy(immutable=True)
G16 = Graph([(1,2),(2,3),(3,4),(4,5),(5,1),(1,3),(1,4)]).canonical_label().copy(immutable=True)
G17 = Graph([(1,2),(2,3),(3,4),(4,5),(5,1),(1,3),(2,4)]).canonical_label().copy(immutable=True)
G19 = Graph([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4),(1,5)]).canonical_label().copy(immutable=True)
G20 = Graph([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4),(1,5),(2,5)]).canonical_label().copy(immutable=True)

Ramsey_Hendry = [(G8, 9), (G9, 9), (G10, 9), (G12, 9), (G13, 10), (G14, 10), (G15, 9), (G16, 10), (G17, 10), (G19, 18), (G20, 18)]

for G, v in Ramsey_Hendry:
    add_to_dict(G, v)

# table IVa and IVb
Ramsey_bipartite = dict()
Ramsey_bipartite[3, 3] = 18
Ramsey_bipartite[2, 2] = 6
Ramsey_bipartite[2, 3] = 10
Ramsey_bipartite[2, 4] = 14
Ramsey_bipartite[2, 5] = 18
Ramsey_bipartite[2, 6] = 21
Ramsey_bipartite[2, 7] = 26
Ramsey_bipartite[2, 8] = 30
Ramsey_bipartite[2, 9] = 33
Ramsey_bipartite[2, 10] = 38
Ramsey_bipartite[2, 11] = 42
Ramsey_bipartite[2, 12] = 46
Ramsey_bipartite[2, 13] = 50
Ramsey_bipartite[2, 14] = 54
Ramsey_bipartite[2, 15] = 57
Ramsey_bipartite[2, 16] = 62

for (n,m), v in Ramsey_bipartite.items():
    add_to_dict(graphs.CompleteBipartiteGraph(n,m), v)

# section 3.3.2 a
def Ramsey_star(n):
    m = n-1
    assert m > 0
    if is_even(m):
        return 2*m-1
    else:
        return 2*m

for n in range(2, N_vertices+1):
    add_to_dict(graphs.StarGraph(n-1), Ramsey_star(n))

# section 3.3.2, n
def Ramsey_m_4_star(n):
    assert n % 4 == 0, "The number of vertices, %s, should be divisible by 4"%n
    m = n//4
    assert m >= 2, "There must be at least 2 copies of the star, but there are only %s"%m
    return 5*m-1

for n in range(8, N_vertices+1, 4):
    add_to_dict(sum([graphs.StarGraph(3) for k in range(n//4)], Graph()), Ramsey_m_4_star(n))

# section 4.1, a,b,c
def Ramsey_cycle(n):
    assert n > 2
    if is_odd(n) and n > 3:
        return 2*n-1
    elif is_even(n) and n > 4:
        return n-1+(n//2)
    elif n == 3 or n == 4:
        return 6

for n in range(3, N_vertices+1):
    add_to_dict(graphs.CycleGraph(n), Ramsey_cycle(n))

# section 4.1, e
def Ramsey_m_triangles(n):
    assert n % 3 == 0, "The number of vertices, %s, should be divisible by 3"%n
    m = n // 3
    assert m >= 2, "There must be at least 2 copies of the triangle, but there are only %s"%m
    return 5*m

for n in range(6, N_vertices+1, 3):
    add_to_dict(sum([graphs.CycleGraph(3) for k in range(n//3)], Graph()), Ramsey_m_triangles(n))

# section 4.1, f
def Ramsey_m_squares(n):
    assert n % 4 == 0, "The number of vertices, %s, should be divisible by 4"%n
    m = n // 4
    assert m >= 2, "There must be at least 2 copies of the square, but there are only %s"%m
    return 6*m-1

for n in range(8, N_vertices+1, 4):
    add_to_dict(sum([graphs.CycleGraph(4) for k in range(n//4)], Graph()), Ramsey_m_squares(n))

# section 5.1
def Ramsey_paths(m):
    assert m >= 2
    return m + (m//2) - 1

for n in range(2, N_vertices+1):
    add_to_dict(graphs.PathGraph(n), Ramsey_paths(n))

# section 5.13 b
def Ramsey_union_K2(m):
    assert is_even(m) and m >= 2
    return 3*(m//2) - 1

for n in range(2, N_vertices+1, 2):
    add_to_dict(sum([graphs.CompleteGraph(2) for k in range(n//2)], Graph()), Ramsey_union_K2(n))

Ramsey_wheel = dict()
# table VIII
Ramsey_wheel[3] = 6
Ramsey_wheel[4] = 18
Ramsey_wheel[5] = 15
Ramsey_wheel[6] = 17
Ramsey_wheel[7] = 19

for n, v in Ramsey_wheel.items():
    add_to_dict(graphs.WheelGraph(n), v)

Ramsey_book = dict()
# table IX
Ramsey_book[1] = 6
Ramsey_book[2] = 10
Ramsey_book[3] = 14
Ramsey_book[4] = 18
Ramsey_book[5] = 21
Ramsey_book[6] = 26

for n, v in Ramsey_book.items():
    add_to_dict(Graph(1).join(graphs.StarGraph(n)), v)

# R.J. Faudree and R.H. Schelp, Ramsey Numbers for All Linear Forests, Discrete Mathematics, 16 (1976) 149-155.

def Ramsey_linear_forest(n, j):
    assert is_even(n-j)
    return n + (n-j)//2 - 1

for n in range(2, N_vertices+1):
    for la in Partitions(n):
        if min(la) > 1:
            G = sum([graphs.PathGraph(p) for p in la], Graph())
            add_to_dict(G, Ramsey_linear_forest(n, sum(1 for p in la if is_odd(p))))

# J.W. Grossman, The Ramsey Numbers of the Union of Two Stars, Utilitas Mathematica, 16 (1979) 271-279.

def Ramsey_two_stars(n, m):
    assert n >= m, "Ramsey_two_stars only valid for n >= m, but n = %s and m = %s"%(n,m)
    return max(n+2*m, 2*n+1, n+m+3)

for n in range(2, N_vertices+1):
    for m in range(2,(n-1)//2+1):
        G = graphs.StarGraph(m-1) + graphs.StarGraph(n-m-1)
        add_to_dict(G, Ramsey_two_stars(n-m-1, m-1))

# Yu, P., & Li, Y. (2016). All Ramsey numbers for brooms in graphs. The Electronic Journal of Combinatorics, 23(3), 3-29.
def Ramsey_broom(k, l):
    assert k >= 2
    n = k+l
    if l == 1 or l == 2:
        return Ramsey_star(n)
    if l == 3:
        return Ramsey_star(n-1)
    if l >= 2*k - 1:
        return n + (l+1)//2 - 1
    if 4 <= l <= 2*k - 2:
        return 2*n - 2*((l+1)//2) - 1

def BroomGraph(k, l):
    G = graphs.StarGraph(k)
    assert G.degree(0) == k
    G.add_path([0] + [k+1+i for i in range(l-1)])
    G.layout("tree", save_pos=True)
    return G

for n in range(2, N_vertices+1):
    for k in range(2,n-1):
        G = BroomGraph(k, n-k)
        add_to_dict(G, Ramsey_broom(k, n-k))

# Jerrold W. Grossman, Frank Harary, Maria Klawe, Generalized Ramsey theory for graphs, X: double stars
        
def Ramsey_double_star(k, l):
    assert k >= 2
    n = k+l+2
    if is_odd(k) and l <= 2:
        return max(2*k+1, k+2*l+2)
    if (is_even(k) or l >= 3) and (k^2 <= 2*l or k >= 3*l):
        return max(2*k+2, k+2*l+2)
        
def DoubleStarGraph(k, l):
    assert k >= l, "DoubleStarGraph only defined for k >= l"
    G = graphs.StarGraph(k)
    H = G.disjoint_union(graphs.StarGraph(l), "pairs")
    H.add_edge((0,0),(1,0))
    return H

for n in range(2, N_vertices+1):
    for l in range(2,n//2):
        k = n-l-2
        G = DoubleStarGraph(k, l)
        assert G.num_verts() == n
        if (is_odd(k) and l <= 2) or ((is_even(k) or l >= 3) and (k^2 <= 2*l or k >= 3*l)):
            add_to_dict(G, Ramsey_double_star(k, l))


######################################################################
# finally, add isolated vertices
for G, v in list(statistic_dict.items()):
    for k in range(N_vertices-G.num_verts()):
        add_to_dict(G.disjoint_union(Graph(k)), max(G.num_verts()+k, v))

N_vertices = 13 # the maximal number of vertices we consider
N_Ramsey = 7 # all graphs with Ramsey number at most N_Ramsey
statistic_dict = dict()
def statistic(G):
    return statistic_dict.get(G.canonical_label().copy(immutable=True))

"""
The Ramsey number of a graph.

This is the smallest integer n such that every two-colouring of the
n vertices of the complete graph K_n contains a (not necessarily
induced) monochromatic copy of the given graph. [1]

Thus, the Ramsey number of the complete graph K_n is the ordinary Ramsey number R(n,n).  Very few of these numbers are known. [2,3]

[1] Chvatál,Rödl,Szemerédi,Trotter, The Ramsey number of a graph with bounded maximum degree. doi:10.1016/0095-8956(83)90037-0

[2] Radziszowski, Stanisław P. "Small Ramsey numbers." Electron. J. Combin 1.7 (1994).

[3] [[wikipedia:Ramsey's theorem#Ramsey numbers]]

[4] Hendry, G. R. T. Ramsey numbers for graphs with five vertices

"""
def check_Ramsey(n, already_found=[]):
    r"""
    Colour the complete graph on n vertices with two colours, and
    return the subgraphs which appear in all colourings.

    EXAMPLES::

        sage: L = dict()
        sage: n = 2; L[n] = check_Ramsey(n)
        sage: n = 3; L[n] = check_Ramsey(n, sum(L.values(), []))
        sage: n = 4; L[n] = check_Ramsey(n, sum(L.values(), []))
        sage: n = 5; L[n] = check_Ramsey(n, sum(L.values(), []))
        sage: n = 6; L[n] = check_Ramsey(n, sum(L.values(), []))
        sage: n = 7; L[n] = check_Ramsey(n, sum(L.values(), []))

        sage: n=6; graphics_array([H.plot() for H in L[n] if H.is_connected()])
    """
    def has_copy(H1, H2, H):
        r"""
        Return True if there is a copy of H in H1=s or H2=E\s.
        """
        S1 = H1.subgraph_search(H)
        if not S1 is None:
            return True
        S2 = H2.subgraph_search(H)
        if not S2 is None:
            return True
        return False

    candidates = [H.canonical_label() for k in range(n+1) for H in graphs(k)]
    candidates = [H for H in candidates if H not in already_found]
    G = graphs.CompleteGraph(n)
    E = Set(G.edges(labels=False))
    V = G.vertices()
    for size in range(1+len(E)//2):
        print("check subgraphs of size", size)
        tested = []
        for s in E.subsets(size):
            H1 = Graph(n)
            H1.add_edges(s)
            H1 = H1.canonical_label()
            H2 = Graph(n)
            H2.add_edges(E.difference(s))
            H2 = H2.canonical_label()
            if (H1, H2) not in tested:
                tested += [(H1, H2)]
                candidates = [H for H in candidates if has_copy(H1, H2, H)]

    return candidates

def add_to_dict(G, v):
    H = G.canonical_label().copy(immutable=True)
    if H in statistic_dict:
        assert statistic_dict[H] == v, "The graph %s should have Ramsey number %s, got %s instead"%(repr(H), statistic_dict[H], v)
        print("%s already known to have Ramsey value %s"%(repr(H), v))
    else:
        statistic_dict[H] = v

Ramsey_small = dict()
for n in range(N_Ramsey+1):
    Ramsey_small[n] = check_Ramsey(n, sum(Ramsey_small.values(), []))

for v, lG in Ramsey_small.items():
    for G in lG:
        add_to_dict(G, v)

# table IIIa
Ramsey_almost_complete = dict()
Ramsey_almost_complete[3] = 3
Ramsey_almost_complete[4] = 10
Ramsey_almost_complete[5] = 22

for n, v in Ramsey_almost_complete.items():
    G = graphs.CompleteGraph(n)
    G.delete_edge(G.edges()[0])
    add_to_dict(G, v)

# G8 - G20 are from Table 1 in
# Hendry, G. R. T. Ramsey numbers for graphs with five vertices

G8 = Graph([(1,2),(2,3),(3,4),(4,5),(3,5)]).canonical_label().copy(immutable=True)
G9 = Graph([(1,2),(2,3),(3,4),(4,5),(2,4)]).canonical_label().copy(immutable=True)
G10 = Graph([(1,2),(1,3),(1,4),(1,5),(2,3)]).canonical_label().copy(immutable=True)
G12 = Graph([(1,2),(2,3),(3,4),(4,5),(1,3),(3,5)]).canonical_label().copy(immutable=True)
G13 = Graph([(1,2),(1,3),(1,4),(1,5),(2,3),(3,4)]).canonical_label().copy(immutable=True)
G14 = Graph([(1,2),(1,3),(1,4),(2,5),(2,3),(3,4)]).canonical_label().copy(immutable=True)
G15 = Graph([(1,2),(2,3),(3,4),(4,5),(5,1),(1,3)]).canonical_label().copy(immutable=True)
G16 = Graph([(1,2),(2,3),(3,4),(4,5),(5,1),(1,3),(1,4)]).canonical_label().copy(immutable=True)
G17 = Graph([(1,2),(2,3),(3,4),(4,5),(5,1),(1,3),(2,4)]).canonical_label().copy(immutable=True)
G19 = Graph([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4),(1,5)]).canonical_label().copy(immutable=True)
G20 = Graph([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4),(1,5),(2,5)]).canonical_label().copy(immutable=True)

Ramsey_Hendry = [(G8, 9), (G9, 9), (G10, 9), (G12, 9), (G13, 10), (G14, 10), (G15, 9), (G16, 10), (G17, 10), (G19, 18), (G20, 18)]

for G, v in Ramsey_Hendry:
    add_to_dict(G, v)

# table IVa and IVb
Ramsey_bipartite = dict()
Ramsey_bipartite[3, 3] = 18
Ramsey_bipartite[2, 2] = 6
Ramsey_bipartite[2, 3] = 10
Ramsey_bipartite[2, 4] = 14
Ramsey_bipartite[2, 5] = 18
Ramsey_bipartite[2, 6] = 21
Ramsey_bipartite[2, 7] = 26
Ramsey_bipartite[2, 8] = 30
Ramsey_bipartite[2, 9] = 33
Ramsey_bipartite[2, 10] = 38
Ramsey_bipartite[2, 11] = 42
Ramsey_bipartite[2, 12] = 46
Ramsey_bipartite[2, 13] = 50
Ramsey_bipartite[2, 14] = 54
Ramsey_bipartite[2, 15] = 57
Ramsey_bipartite[2, 16] = 62

for (n,m), v in Ramsey_bipartite.items():
    add_to_dict(graphs.CompleteBipartiteGraph(n,m), v)

# section 3.3.2 a
def Ramsey_star(n):
    m = n-1
    assert m > 0
    if is_even(m):
        return 2*m-1
    else:
        return 2*m

for n in range(2, N_vertices+1):
    add_to_dict(graphs.StarGraph(n-1), Ramsey_star(n))

# section 3.3.2, n
def Ramsey_m_4_star(n):
    assert n % 4 == 0, "The number of vertices, %s, should be divisible by 4"%n
    m = n//4
    assert m >= 2, "There must be at least 2 copies of the star, but there are only %s"%m
    return 5*m-1

for n in range(8, N_vertices+1, 4):
    add_to_dict(sum([graphs.StarGraph(3) for k in range(n//4)], Graph()), Ramsey_m_4_star(n))

# section 4.1, a,b,c
def Ramsey_cycle(n):
    assert n > 2
    if is_odd(n) and n > 3:
        return 2*n-1
    elif is_even(n) and n > 4:
        return n-1+(n//2)
    elif n == 3 or n == 4:
        return 6

for n in range(3, N_vertices+1):
    add_to_dict(graphs.CycleGraph(n), Ramsey_cycle(n))

# section 4.1, e
def Ramsey_m_triangles(n):
    assert n % 3 == 0, "The number of vertices, %s, should be divisible by 3"%n
    m = n // 3
    assert m >= 2, "There must be at least 2 copies of the triangle, but there are only %s"%m
    return 5*m

for n in range(6, N_vertices+1, 3):
    add_to_dict(sum([graphs.CycleGraph(3) for k in range(n//3)], Graph()), Ramsey_m_triangles(n))

# section 4.1, f
def Ramsey_m_squares(n):
    assert n % 4 == 0, "The number of vertices, %s, should be divisible by 4"%n
    m = n // 4
    assert m >= 2, "There must be at least 2 copies of the square, but there are only %s"%m
    return 6*m-1

for n in range(8, N_vertices+1, 4):
    add_to_dict(sum([graphs.CycleGraph(4) for k in range(n//4)], Graph()), Ramsey_m_squares(n))

# section 5.1
def Ramsey_paths(m):
    assert m >= 2
    return m + (m//2) - 1

for n in range(2, N_vertices+1):
    add_to_dict(graphs.PathGraph(n), Ramsey_paths(n))

# section 5.13 b
def Ramsey_union_K2(m):
    assert is_even(m) and m >= 2
    return 3*(m//2) - 1

for n in range(2, N_vertices+1, 2):
    add_to_dict(sum([graphs.CompleteGraph(2) for k in range(n//2)], Graph()), Ramsey_union_K2(n))

Ramsey_wheel = dict()
# table VIII
Ramsey_wheel[3] = 6
Ramsey_wheel[4] = 18
Ramsey_wheel[5] = 15
Ramsey_wheel[6] = 17
Ramsey_wheel[7] = 19

for n, v in Ramsey_wheel.items():
    add_to_dict(graphs.WheelGraph(n), v)

Ramsey_book = dict()
# table IX
Ramsey_book[1] = 6
Ramsey_book[2] = 10
Ramsey_book[3] = 14
Ramsey_book[4] = 18
Ramsey_book[5] = 21
Ramsey_book[6] = 26

for n, v in Ramsey_book.items():
    add_to_dict(Graph(1).join(graphs.StarGraph(n)), v)

# R.J. Faudree and R.H. Schelp, Ramsey Numbers for All Linear Forests, Discrete Mathematics, 16 (1976) 149-155.

def Ramsey_linear_forest(n, j):
    assert is_even(n-j)
    return n + (n-j)//2 - 1

for n in range(2, N_vertices+1):
    for la in Partitions(n):
        if min(la) > 1:
            G = sum([graphs.PathGraph(p) for p in la], Graph())
            add_to_dict(G, Ramsey_linear_forest(n, sum(1 for p in la if is_odd(p))))

# J.W. Grossman, The Ramsey Numbers of the Union of Two Stars, Utilitas Mathematica, 16 (1979) 271-279.

def Ramsey_two_stars(n, m):
    assert n >= m, "Ramsey_two_stars only valid for n >= m, but n = %s and m = %s"%(n,m)
    return max(n+2*m, 2*n+1, n+m+3)

for n in range(2, N_vertices+1):
    for m in range(2,(n-1)//2+1):
        G = graphs.StarGraph(m-1) + graphs.StarGraph(n-m-1)
        add_to_dict(G, Ramsey_two_stars(n-m-1, m-1))

# Yu, P., & Li, Y. (2016). All Ramsey numbers for brooms in graphs. The Electronic Journal of Combinatorics, 23(3), 3-29.
def Ramsey_broom(k, l):
    assert k >= 2
    n = k+l
    if l == 1 or l == 2:
        return Ramsey_star(n)
    if l == 3:
        return Ramsey_star(n-1)
    if l >= 2*k - 1:
        return n + (l+1)//2 - 1
    if 4 <= l <= 2*k - 2:
        return 2*n - 2*((l+1)//2) - 1

def BroomGraph(k, l):
    G = graphs.StarGraph(k)
    assert G.degree(0) == k
    G.add_path([0] + [k+1+i for i in range(l-1)])
    G.layout("tree", save_pos=True)
    return G

for n in range(2, N_vertices+1):
    for k in range(2,n-1):
        G = BroomGraph(k, n-k)
        add_to_dict(G, Ramsey_broom(k, n-k))

# Jerrold W. Grossman, Frank Harary, Maria Klawe, Generalized Ramsey theory for graphs, X: double stars
        
def Ramsey_double_star(k, l):
    assert k >= 2
    n = k+l+2
    if is_odd(k) and l <= 2:
        return max(2*k+1, k+2*l+2)
    if (is_even(k) or l >= 3) and (k^2 <= 2*l or k >= 3*l):
        return max(2*k+2, k+2*l+2)
        
def DoubleStarGraph(k, l):
    assert k >= l, "DoubleStarGraph only defined for k >= l"
    G = graphs.StarGraph(k)
    H = G.disjoint_union(graphs.StarGraph(l), "pairs")
    H.add_edge((0,0),(1,0))
    return H

for n in range(2, N_vertices+1):
    for l in range(2,n//2):
        k = n-l-2
        G = DoubleStarGraph(k, l)
        assert G.num_verts() == n
        if (is_odd(k) and l <= 2) or ((is_even(k) or l >= 3) and (k^2 <= 2*l or k >= 3*l)):
            add_to_dict(G, Ramsey_double_star(k, l))


######################################################################
# finally, add isolated vertices
for G, v in list(statistic_dict.items()):
    for k in range(N_vertices-G.num_verts()):
        add_to_dict(G.disjoint_union(Graph(k)), max(G.num_verts()+k, v))

def statistic(pi):
    i = 1
    pos = 0
    count = 0
    while i < len(pi):
        if pi[pos] == i:
            i += 1
        elif pi[pos] > i:
            count += 1
        pos = (pos+1)%len(pi)
    return count
def statistic(pi):
    n = len(pi)
    pinv = pi.inverse()
    return sum( n-i for i in [1 .. n-1] if pinv(i) > pinv(i+1) )

def statistic(pi):
    i = 1
    pos = 0
    count = 0
    while i < len(pi):
        if pi[pos] == i:
            i += 1
        elif pi[pos] > i:
            count += 1
        pos = (pos+1)%len(pi)
    return count
def statistic(pi):
    n = len(pi)
    pinv = pi.inverse()
    return sum( n-i for i in [1 .. n-1] if pinv(i) > pinv(i+1) )

def statistic(x):
    xw = 1
    lvl = 1
    for step in x:
        if step == 0:
            lvl += 1
        else:
            xw *= lvl
    return xw

def dyck_box_weightstatistic(x):
    xw = 1
    lvl = 1
    for step in x:
        if step == 0:
            lvl += 1
        else:
            xw *= lvl
    return xw

def statistic(p):
    return sum( i+1 for i in range(len(p)-1) if p[i] == p[i+1] == 0 )

def count_prime_leaves(m):
    from sage.graphs.graph_decompositions.modular_decomposition import NodeType
    if m.label() == NodeType.PRIME:
        return len(m.leaf_labels())
    return sum(count_prime_leaves(c) for c in m)

def statistic(G):
    m = G.modular_decomposition(algorithm='habib', style='tree')
    return count_prime_leaves(m)

def count_prime_leaves(m):
    from sage.graphs.graph_decompositions.modular_decomposition import NodeType
def count_prime_leaves(m):
    if m.label() == NodeType.PRIME:
        return len(m.leaf_labels())
    return sum(count_prime_leaves(c) for c in m)

def statistic(G):
    m = G.modular_decomposition(algorithm='habib', style='tree')
    return count_prime_leaves(m)

def statistic(G):
    """Return the smallest number such that Alice has a winning strategy.

    EXAMPLES:

    From http://www.dlsu.edu.ph/conferences/dlsu_research_congress/2014/_pdf/proceedings/TPHS-I-011-FT.pdf::

        sage: [statistic(graphs.PathGraph(r)) for r in range(2,12)]
        [2, 2, 3, 3, 3, 3, 3, 3, 3, 3]
    
        sage: [statistic(graphs.CycleGraph(r)) for r in range(2,10)]
        [2, 3, 3, 3, 3, 3, 3, 3]
    
        sage: [statistic(graphs.StarGraph(r)) for r in range(2,10)]
        [2, 2, 2, 2, 2, 2, 2, 2]
    
        sage: [statistic(graphs.CompleteGraph(r)) for r in range(2,10)]
        [2, 3, 4, 5, 6, 7, 8, 9]

    Proposition 4 of the reference seems to miss some corner cases.
    W(3) and W(4) are complete graphs, so Proposition 5 applies.  

    However, W(5) and W(7) seem to be special::

        sage: [statistic(graphs.WheelGraph(r)) for r in range(5,12)] 
        [3, 4, 3, 4, 4, 4, 4]
    
        sage: [[statistic(graphs.CompleteBipartiteGraph(r, s)) for r in range(1,6)] for s in range(1,6)]
        [[2, 2, 2, 2, 2],
        [2, 3, 3, 3, 3],
        [2, 3, 3, 3, 3],
        [2, 3, 3, 3, 3],
        [2, 3, 3, 3, 3]]

    The following contradicts Theorem 1, but is in agreement with
    http://ssltest.cs.umd.edu/~gasarch/TOPICS/graphcolgame/tree3.pdf,
    Theorem 3::

        sage: statistic(graphs.PetersenGraph())
        4

    from wikipedia::

        sage: [statistic(graphs.CompleteGraph(2).cartesian_product(graphs.PathGraph(r))) for r in range(1,6)]
        [2, 3, 3, 4, 4]
    
        sage: [statistic(graphs.CompleteGraph(2).cartesian_product(graphs.CycleGraph(r))) for r in range(1,5)]
        [2, 3, 4, 4]
    
        sage: [statistic(graphs.CompleteGraph(2).cartesian_product(graphs.CompleteGraph(r))) for r in range(1,5)]
        [2, 3, 4, 5]

    """
    G = G.relabel(immutable=True, inplace=False)
    for k in range(G.chromatic_number(), max(G.degree())+2):
        if Alice_wins(G, k):
            return k

def normalize_colours(D):
    """
    From left to right, use small colours first.
    """
    pi = [None]*(max(c for c in D if c is not None)+1)
    i = 0
    for e in D:
        if e is not None and pi[e] is None:
            pi[e] = i
            i += 1
    return tuple([None if e is None else pi[e] for e in D])

@cached_function
def Alice_wins(G, k, C=None):
    """Return whether C is a winning position for Alice, who wants to
    properly color the graph.

    Expect that the vertices are labelled 0,1,...

    """
    def children(D):
        if all(c is None for c in D):
            colours = 1
        else:
            colours = min(k, max(c for c in D if c is not None) + 2)
        for v, c in enumerate(D):
            if c is None:
                for d in range(colours):
                    if not any(D[u] == d for u in G.neighbor_iterator(v)):
                        yield D[:v] + tuple([d]) + D[v+1:]

    if C is None:
        C = tuple([None]*G.num_verts())

    uncoloured_vertices = C.count(None)
    if uncoloured_vertices == 0:
        return True

    # let Alice colour a vertex
    for D in children(C):
        # if C was missing precisely one vertex, and we could
        # colour it, Alice wins
        if uncoloured_vertices == 1:
            return True
        # Alice wins if there is a D such that all moves of Bob make Alice win
        has_colouring = False
        for E in children(D):
            has_colouring = True
            if not Alice_wins(G, k, normalize_colours(E)):
                break
        else:
            if has_colouring:
                return True
    return False

def statistic(G):
    """Return the smallest number such that Alice has a winning strategy.

    EXAMPLES:

    From http://www.dlsu.edu.ph/conferences/dlsu_research_congress/2014/_pdf/proceedings/TPHS-I-011-FT.pdf::

        sage: [statistic(graphs.PathGraph(r)) for r in range(2,12)]
        [2, 2, 3, 3, 3, 3, 3, 3, 3, 3]
    
        sage: [statistic(graphs.CycleGraph(r)) for r in range(2,10)]
        [2, 3, 3, 3, 3, 3, 3, 3]
    
        sage: [statistic(graphs.StarGraph(r)) for r in range(2,10)]
        [2, 2, 2, 2, 2, 2, 2, 2]
    
        sage: [statistic(graphs.CompleteGraph(r)) for r in range(2,10)]
        [2, 3, 4, 5, 6, 7, 8, 9]

    Proposition 4 of the reference seems to miss some corner cases.
    W(3) and W(4) are complete graphs, so Proposition 5 applies.  

    However, W(5) and W(7) seem to be special::

        sage: [statistic(graphs.WheelGraph(r)) for r in range(5,12)] 
        [3, 4, 3, 4, 4, 4, 4]
    
        sage: [[statistic(graphs.CompleteBipartiteGraph(r, s)) for r in range(1,6)] for s in range(1,6)]
        [[2, 2, 2, 2, 2],
        [2, 3, 3, 3, 3],
        [2, 3, 3, 3, 3],
        [2, 3, 3, 3, 3],
        [2, 3, 3, 3, 3]]

    The following contradicts Theorem 1, but is in agreement with
    http://ssltest.cs.umd.edu/~gasarch/TOPICS/graphcolgame/tree3.pdf,
    Theorem 3::

        sage: statistic(graphs.PetersenGraph())
        4

    from wikipedia::

        sage: [statistic(graphs.CompleteGraph(2).cartesian_product(graphs.PathGraph(r))) for r in range(1,6)]
        [2, 3, 3, 4, 4]
    
        sage: [statistic(graphs.CompleteGraph(2).cartesian_product(graphs.CycleGraph(r))) for r in range(1,5)]
        [2, 3, 4, 4]
    
        sage: [statistic(graphs.CompleteGraph(2).cartesian_product(graphs.CompleteGraph(r))) for r in range(1,5)]
        [2, 3, 4, 5]

    """
    G = G.relabel(immutable=True, inplace=False)
    for k in range(G.chromatic_number(), max(G.degree())+2):
        if Alice_wins(G, k):
            return k

def normalize_colours(D):
    """
    From left to right, use small colours first.
    """
    pi = [None]*(max(Dc for c in D if c is not None)+1)
    i = 0
    for e in D:
        if e is not None and pi[e] is None:
            pi[e] = i
            i += 1
    return tuple([None if e is None else pi[e] for e in D])

@cached_function
def Alice_wins(G, k, C=None):
    """Return whether C is a winning position for Alice, who wants to
    properly color the graph.

    Expect that the vertices are labelled 0,1,...

    """
    def children(D):
        if all(c is None for c in D):
            colours = 1
        else:
            colours = min(k, max(D)+c for c in D if c is not None) + 2)
        for v, c in enumerate(D):
            if c is None:
                for d in range(colours):
                    if not any(D[u] == d for u in G.neighbor_iterator(v)):
                        yield D[:v] + tuple([d]) + D[v+1:]

    if C is None:
        C = tuple([None]*G.num_verts())

    uncoloured_vertices = C.count(None)
    if uncoloured_vertices == 0:
        return True

    # let Alice colour a vertex
    for D in children(C):
        # if C was missing precisely one vertex, and we could
        # colour it, Alice wins
        if uncoloured_vertices == 1:
            return True
        # Alice wins if there is a D such that all moves of Bob make Alice win
        has_colouring = False
        for E in children(D):
            has_colouring = True
            if not Alice_wins(G, k, normalize_colours(E)):
                break
        else:
            if has_colouring:
                return True
    return False

def is_dominating(G,V):        
    return set(G.vertices()) == set(V).union(*[G.neighbors(v) for v in V])

def minimal_dominating_sets(G):
    Vs = list(Subsets(G.vertices()))
    mdoms = []
    while Vs:
        V = Vs.pop(0)
        if is_dominating(G,V):
            mdoms.append(V)
            Vs = list(filter(lambda X: not X.issuperset(V), Vs))
    return mdoms

def statistic(G):
    return len(minimal_dominating_sets(G))

def is_dominating(G,V):        
    return set(G.vertices()) == set(V).union(*[G.neighbors(v) for v in V])

def minimal_dominating_sets(G):
    Vs = list(Subsets(G.vertices()))
    mdoms = []
    while Vs:
        V = Vs.pop(0)
        if is_dominating(G,V):
            mdoms.append(V)
            Vs = list(filter( lambda X: not X.issuperset(V), Vs))
    return mdoms

def statistic(G):
    return len(minimal_dominating_sets(G))

def statistic(g):
    return sum(h.szeged_index() for h in g.connected_components_subgraphs())

def statistic(g):
    return gsum(h.szeged_index() for h in g.connected_components_subgraphs())

def statistic(C):
    from sage.combinat.q_analogues import q_int
    q = polygen(ZZ, 'q')
    return max(prod(q_int(d, q) for d in WeylGroup(C).degrees()).list())

def statistic(C):
    from sage.combinat.q_analogues import q_int
    q = polygen(ZZ, 'q')
    return max(prod(q_int(d, q) for d in WeylGroup(C).degrees()).list())

def filling(w):
    w = list(w)
    f = list(w)
    for i in range(len(w)-1):
        if w[i:i+2] == [0, 1]:
            f[i:i+2] = [1, 0]
    if w[len(w)-1] == 0:
        f[len(w)-1] = 1
    return Words([0,1])(f)

def statistic(w):
    d = 0
    while w.count(0):
        w = filling(w)
        d += 1
    return d

def filling(w):
    w = list(w)
    f = list(w)
    for i in range(len(w)-1):
        if w[i:i+2] == [0, 1]:
            f[i:i+2] = [1, 0]
    if w[len(w)-1] == 0:
        f[len(w)-1] = 1
    return Words([0,1])(f)

def degreestatistic(w):
    d = 0
    while w.count(0):
        w = filling(w)
        d += 1
    return d

def statistic(D):
    D = DyckWord(D)
    n = len(D)    
    tunnels = D.tunnels()
    t = [(i, j) for (i, j) in tunnels if i + j == n]
    res = 0
    for (a1, b1) in t:
        for (a2, b2) in t:
            if a1 < a2 < b2 < b1:
                res += 1
    return res

def count_pd4_simples(dyck_word):
    c = kupisch(dyck_word)
    n = len(c)

    count = 0
    for i in range(n):
        a, l = i, 1
        pd = 0

        while l < c[a] and pd < 4:
            a, l = a + l, c[a] - l
            pd += 1

        if pd == 4 and l == c[a]:
            count += 1

    return count

def kupisch(D):
    DR = D.reverse()
    H = DR.heights()
    return [1 + H[i] for i, s in enumerate(DR) if s == 0] + [1]

def statistic(D):
    return count_pd4_simples(D)

gap('LoadPackage("QPA");')

import tempfile as _tf, os as _os
_gap_code = r"""
DeclareOperation("numberssimpprojdim4", [IsList]);

InstallMethod(numberssimpprojdim4, "for a representation of a quiver", [IsList],0,function(L)
    local A, R, RR, list;
    list := L;

    A := NakayamaAlgebra(GF(3),list);
    R := SimpleModules(A);
    RR := Filtered(R,x->ProjDimensionOfModule(x,4)=4);
    return(Size(RR));
end
);
"""
with _tf.NamedTemporaryFile(mode="w", suffix=".g", delete=False, dir="/tmp") as _f:
    _f.write('LoadPackage("QPA");;\n')
    _f.write(_gap_code)
    _tmp = _f.name
gap.eval('Read("' + _tmp + '");')
_os.unlink(_tmp)def count_pd4_simples(dyck_word):
    c = kupisch(dyck_word)
    n = len(c)

    count = 0
    for i in range(n):
        a, l = i, 1
        pd = 0

        while l < c[a] and pd < 4:
            a, l = a + l, c[a] - l
            pd += 1

        if pd == 4 and l == c[a]:
            count += 1

    return count

def kupisch(D):
    DR = D.reverse()
    H = DR.heights()
    return [1 + H[i] for i, s in enumerate(DR) if s == 0] + [1]

def statistic(D):
    K = kupisch(D)
    return ZZ(gap.numberssimpprojdim4(K)return count_pd4_simples(D)