Intercalates and discrepancy in random Latin squares. Issue 2 (8th November 2017) - Kwan, Matthew; Sudakov, Benny
Abstract: An intercalate in a Latin square is a 2 × 2 Latin subsquare. Let N be the number of intercalates in a uniformly random n × n Latin square. We prove that asymptotically almost surely N ≥ ( 1 − o ( 1 ) )   n 2 / 4, and that E N ≤ ( 1 + o ( 1 ) )   n 2 / 2 (therefore asymptotically almost surely N ≤ f n 2 for any f → ∞ ). This significantly improves the previous best lower and upper bounds. We also give an upper tail bound for the number of intercalates in 2 fixed rows of a random Latin square. In addition, we discuss a problem of Linial and Luria on low‐discrepancy Latin squares.