Leah Howard and Wendy Myrvold (University of Victoria, 2009) have found the
first known counterexample to my general conjecture concerning p-ranks
of nets. They in fact found a 4-net of order 10 having 2-rank 34, yet whose 3-subnets have 2-rank equal to 28.
This means a gap in 2-rank (between the 4-net and each 3-subnet) of only 6 (one less than
the conjectured minimum value of 7). The example, which has been submitted for
publication in the Bulletin of the ICA, is specified by the following pair
of orthogonal Latin squares of order 10:
|
00
|
67
|
58
|
49
|
83
|
72
|
91
|
16
|
25
|
34
|
|
76
|
11
|
84
|
57
|
29
|
93
|
08
|
35
|
60
|
42
|
|
85
|
48
|
22
|
96
|
71
|
09
|
37
|
63
|
14
|
50
|
|
94
|
75
|
69
|
33
|
07
|
18
|
82
|
40
|
51
|
26
|
|
38
|
92
|
17
|
70
|
44
|
86
|
59
|
21
|
03
|
65
|
|
27
|
39
|
90
|
81
|
68
|
55
|
74
|
02
|
46
|
13
|
|
19
|
80
|
73
|
28
|
95
|
47
|
66
|
54
|
32
|
01
|
|
61
|
53
|
36
|
04
|
12
|
20
|
45
|
77
|
88
|
99
|
|
43
|
24
|
05
|
62
|
56
|
31
|
10
|
89
|
97
|
78
|
|
52
|
06
|
41
|
15
|
30
|
64
|
23
|
98
|
79
|
87
|
This 4-net does not extend to a 5-net.
Lately I have been focusing on nets of prime order only. No counterexamples are yet known in this case, but since seeing the Howard-Myrvold example, I won't be surprised if it is only a matter of time before such a counterexample is found.
I am optimistic that the conjecture holds for nets which are completable to affine planes (as in the revised formulation below). Even this special case of the conjecture has all the merits of the original conjecture (its validity would imply that projective planes of squarefree order must in fact be Desarguesian of prime order). But to resolve this special case will require one of two things: either a proof of the validity of the revised conjecture, or a projective plane of non-prime power order. Either way, a very tall order!
For the record, I state here the revised conjecture:
Conjecture. Let N be a k-net of order n, and assume p is a prime sharply dividing n (i.e. n is divisible by p but not by p2). Assume that N is completable to an (n+1)-net, i.e. an affine plane of order n. Then for every (k-1)-subnet N' of N, we have rankp(N)-rankp(N') <= n-k+1.
/ revised December, 2009