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 ( n 3 ) if m ∈ { 2 , 3 , 5 } and ζ ( n , m ) = Θ ( n 4 ) if m ∈ { 4 , 6 , 9 , 10 } . In particular, 1 8 n 3 + O ( n 2 ) ≤ ζ ( n , 2 ) ≤ 1 4 n 3 + O ( n 2 ) and 1 27 n 3 + O ( n 5 / 2 ) ≤ ζ ( n , 3 ) ≤ 1 18 n 3 + O ( n 2 ) for all n. We find an explicit bound on ζ ( n , 2 d ) of the form Θ ( n d + 2 ) and which is achieved only by the elementary abelian 2-groups. 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 ( n 3 ) . For a large class of Latin squares L we show that ζ ⁎ ( n , L ) = O ( n 3 ) . For any Latin square L we give an ϵ in the interval ( 0 , 1 ) such that ζ ⁎ ( n , L ) ≥ Ω ( n 2 + ϵ ) . We believe that this bound is achieved for certain squares L.