-- "Ex. 3.4 in De Loera paper Grobner Bases and Graph Colorings";
R = QQ[a,b,c,d,e,f,g,h,i,j];
-- ideal J(8,4)
I = ideal(
 ( (a-b)*(a-c)*(a-d)*(b-c)*(b-d)*(c-d) ),
 ( (a-b)*(a-c)*(a-e)*(b-c)*(b-e)*(c-e) ),
 ( (a-b)*(a-c)*(a-f)*(b-c)*(b-f)*(c-f) ),
 ( (a-b)*(a-c)*(a-g)*(b-c)*(b-g)*(c-g) ),
 ( (a-b)*(a-c)*(a-h)*(b-c)*(b-h)*(c-h) ),
 ( (a-b)*(a-d)*(a-e)*(b-d)*(b-e)*(d-e) ),
 ( (a-b)*(a-d)*(a-f)*(b-d)*(b-f)*(d-f) ),
 ( (a-b)*(a-d)*(a-g)*(b-d)*(b-g)*(d-g) ),
 ( (a-b)*(a-d)*(a-h)*(b-d)*(b-h)*(d-h) ),
 ( (a-b)*(a-e)*(a-f)*(b-e)*(b-f)*(e-f) ),
 ( (a-b)*(a-e)*(a-g)*(b-e)*(b-g)*(e-g) ),
 ( (a-b)*(a-e)*(a-h)*(b-e)*(b-h)*(e-h) ),
 ( (a-b)*(a-f)*(a-g)*(b-f)*(b-g)*(f-g) ),
 ( (a-b)*(a-f)*(a-h)*(b-f)*(b-h)*(f-h) ),
 ( (a-b)*(a-g)*(a-h)*(b-g)*(b-h)*(g-h) ),
 ( (a-c)*(a-d)*(a-e)*(c-d)*(c-e)*(d-e) ),
 ( (a-c)*(a-d)*(a-f)*(c-d)*(c-f)*(d-f) ),
 ( (a-c)*(a-d)*(a-g)*(c-d)*(c-g)*(d-g) ),
 ( (a-c)*(a-d)*(a-h)*(c-d)*(c-h)*(d-h) ),
 ( (a-c)*(a-e)*(a-f)*(c-e)*(c-f)*(e-f) ),
 ( (a-c)*(a-e)*(a-g)*(c-e)*(c-g)*(e-g) ),
 ( (a-c)*(a-e)*(a-h)*(c-e)*(c-h)*(e-h) ),
 ( (a-c)*(a-f)*(a-g)*(c-f)*(c-g)*(f-g) ),
 ( (a-c)*(a-f)*(a-h)*(c-f)*(c-h)*(f-h) ),
 ( (a-c)*(a-g)*(a-h)*(c-g)*(c-h)*(g-h) ),
 ( (a-d)*(a-e)*(a-f)*(d-e)*(d-f)*(e-f) ),
 ( (a-d)*(a-e)*(a-g)*(d-e)*(d-g)*(e-g) ),
 ( (a-d)*(a-e)*(a-h)*(d-e)*(d-h)*(e-h) ),
 ( (a-d)*(a-f)*(a-g)*(d-f)*(d-g)*(f-g) ),
 ( (a-d)*(a-f)*(a-h)*(d-f)*(d-h)*(f-h) ),
 ( (a-d)*(a-g)*(a-h)*(d-g)*(d-h)*(g-h) ),
 ( (a-e)*(a-f)*(a-g)*(e-f)*(e-g)*(f-g) ),
 ( (a-e)*(a-f)*(a-h)*(e-f)*(e-h)*(f-h) ),
 ( (a-e)*(a-g)*(a-h)*(e-g)*(e-h)*(g-h) ),
 ( (a-f)*(a-g)*(a-h)*(f-g)*(f-h)*(g-h) ),
 ( (b-c)*(b-d)*(b-e)*(c-d)*(c-e)*(d-e) ),
 ( (b-c)*(b-d)*(b-f)*(c-d)*(c-f)*(d-f) ),
 ( (b-c)*(b-d)*(b-g)*(c-d)*(c-g)*(d-g) ),
 ( (b-c)*(b-d)*(b-h)*(c-d)*(c-h)*(d-h) ),
 ( (b-c)*(b-e)*(b-f)*(c-e)*(c-f)*(e-f) ),
 ( (b-c)*(b-e)*(b-g)*(c-e)*(c-g)*(e-g) ),
 ( (b-c)*(b-e)*(b-h)*(c-e)*(c-h)*(e-h) ),
 ( (b-c)*(b-f)*(b-g)*(c-f)*(c-g)*(f-g) ),
 ( (b-c)*(b-f)*(b-h)*(c-f)*(c-h)*(f-h) ),
 ( (b-c)*(b-g)*(b-h)*(c-g)*(c-h)*(g-h) ),
 ( (b-d)*(b-e)*(b-f)*(d-e)*(d-f)*(e-f) ),
 ( (b-d)*(b-e)*(b-g)*(d-e)*(d-g)*(e-g) ),
 ( (b-d)*(b-e)*(b-h)*(d-e)*(d-h)*(e-h) ),
 ( (b-d)*(b-f)*(b-g)*(d-f)*(d-g)*(f-g) ),
 ( (b-d)*(b-f)*(b-h)*(d-f)*(d-h)*(f-h) ),
 ( (b-d)*(b-g)*(b-h)*(d-g)*(d-h)*(g-h) ),
 ( (b-e)*(b-f)*(b-g)*(e-f)*(e-g)*(f-g) ),
 ( (b-e)*(b-f)*(b-h)*(e-f)*(e-h)*(f-h) ),
 ( (b-e)*(b-g)*(b-h)*(e-g)*(e-h)*(g-h) ),
 ( (b-f)*(b-g)*(b-h)*(f-g)*(f-h)*(g-h) ),
 ( (c-d)*(c-e)*(c-f)*(d-e)*(d-f)*(e-f) ),
 ( (c-d)*(c-e)*(c-g)*(d-e)*(d-g)*(e-g) ),
 ( (c-d)*(c-e)*(c-h)*(d-e)*(d-h)*(e-h) ),
 ( (c-d)*(c-f)*(c-g)*(d-f)*(d-g)*(f-g) ),
 ( (c-d)*(c-f)*(c-h)*(d-f)*(d-h)*(f-h) ),
 ( (c-d)*(c-g)*(c-h)*(d-g)*(d-h)*(g-h) ),
 ( (c-e)*(c-f)*(c-g)*(e-f)*(e-g)*(f-g) ),
 ( (c-e)*(c-f)*(c-h)*(e-f)*(e-h)*(f-h) ),
 ( (c-e)*(c-g)*(c-h)*(e-g)*(e-h)*(g-h) ),
 ( (c-f)*(c-g)*(c-h)*(f-g)*(f-h)*(g-h) ),
 ( (d-e)*(d-f)*(d-g)*(e-f)*(e-g)*(f-g) ),
 ( (d-e)*(d-f)*(d-h)*(e-f)*(e-h)*(f-h) ),
 ( (d-e)*(d-g)*(d-h)*(e-g)*(e-h)*(g-h) ),
 ( (d-f)*(d-g)*(d-h)*(f-g)*(f-h)*(g-h) ),
 ( (e-f)*(e-g)*(e-h)*(f-g)*(f-h)*(g-h) )
);
-- PH for 4 x 2 grid
M = ideal( ((a-b)*(b-c)*(c-d)*(d-e)*(e-f)*(f-g)*(g-h)*(a-h)*(b-g)*(c-f)) );
M1 = ideal( (a-b) );
M2 = ideal( (b-c) );
M3 = ideal( (c-d) );
M4 = ideal( (d-e) );
M5 = ideal( (e-f) );
M6 = ideal( (f-g) );
M7 = ideal( (g-h) );
M8 = ideal( (a-h) );
M9 = ideal( (b-g) );
M10 = ideal( (c-f) );
J1 = saturate(I, M1);
J2 = saturate(I, M2);
J3 = saturate(I, M3);
J4 = saturate(I, M4);
J5 = saturate(I, M5);
J6 = saturate(I, M6);
J7 = saturate(I, M7);
J8 = saturate(I, M8);
J9 = saturate(I, M9);
J10 = saturate(I, M10);
--JI = intersect(J1, J2, J3, J4, J5, J6, J7, J8, J9, J10);
--degree(JI)
JU = ideal(gens(J1), gens(J2), gens(J3), gens(J4), gens(J5), gens(J6), gens(J7), gens(J8), gens(J9), gens(J10));
degree(JU)