Proposition

Proposition 2 ([Balanced types and parity]). For all sufficiently large m≡14 , the prescribed sets S_x⊆T , x∈A B , satisfy the following identities. For every Y∈A,B and t∈T , | x∈Y:t∈S_x |=|Y|p. Their intersection cardinalities satisfy …

Proposition 2 ([Balanced types and parity]). For all sufficiently large m≡14 , the prescribed sets Sx⊆T , x∈A B , satisfy the following identities. For every Y∈A,B and t∈T , | x∈Y:t∈Sx |=|Y|p. Their intersection cardinalities satisfy |Sx∩S_ x' |, ≡12 , , (x,x'∈A, (x,x')≠(xA,xA)),; |SxA |, ≡02,; |Sx∩Sy|, ≡12 , , (x∈A, y∈B). Moreover, 129≤|Sx|≤136 at every vertex.

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Proposition 1 ([Parity criterion]). Let Γ_A,Γ_B be finite loopless graphs…Theorem 1.1. There exist a finitely presented torsion-free group G and elements…Proposition
An ordinary letter lies on q+1 lines, so occurs at exactly (q+1)m=np vertices of A and (q+1)(m-1)=(n-v)p vertices of B . An extra letter belongs to four members of D , and therefore occurs 4a_m+1=(q+1)m times in A and 4b_m=(q+1)(m-1) times in B . This proves. The intersection of two ordinary parts has size q+1=129 when the lines agree and size one otherwise. For the extra parts, all intersections have even size except the intersection of the full seven-element set with itself. In fact, the possible even sizes are zero, two, and four. Only x_A has the full extra part, giving exactly the exception in. Finally, the possible sizes of S_x are 129 , 133 , and 136 .
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