## Abstract

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.

Original language | English |
---|---|

Pages (from-to) | 41-56 |

Journal | Journal of Combinatorial Theory, Series A |

Volume | 124 |

DOIs | |

Publication status | Published - May 2014 |

## Keywords

- Latin square
- Subsquare
- Intercalate
- Number of subgroups
- Dihedral group
- Chein loop