Bounds on the number of small Latin subsquares

Let ζ(n, m) be the largest number of order m subsquares
achieved by any Latin square of order n. We show that
ζ(n, m) = Θ(n3 ) if m
∈ {2, 3, 5}
and ζ(n, m) = Θ(n4 ) if
m
∈ {4, 6, 9, 10}. In particular,
1
8
n3 + O(n2 ) . ζ(n, 2) .
1
4
n3 + O(n2 ) and
1
27
n3 + O(n5/2
) . ζ(n, 3) .
1
18
n3 + O(n2 )
for all n. We find an explicit bound on ζ(n, 2d ) of the form
Θ(nd+2
) and which is achieved only by the elementary abelian
2-groups.
For a fixed Latin square L let
ζ∗ (n, L) be the largest number
of subsquares isotopic to L achieved by any Latin square
of order n. When L is a cyclic Latin square we show that
ζ∗ (n, L) = Θ(n3 ). For a large class of Latin squares L we
show that
ζ∗ (n, L) = O(n3 ). For any Latin square L we give
an in the interval (0, 1) such that
ζ∗ (n, L) . Ω(n2+
). We
believe that this bound is achieved for certain squares L.