On constants in the Füredi–Hajnal and the Stanley–Wilf conjecture

Abstract

For a given permutation matrix P, let f P ( n ) be the maximum number of 1-entries in an n × n ( 0 , 1 ) -matrix avoiding P and let S P ( n ) be the set of all n × n permutation matrices avoiding P. The Füredi–Hajnal conjecture asserts that c P : = lim n → ∞ f P ( n ) / n is finite, while the Stanley–Wilf conjecture asserts that s P : = lim n → ∞ | S P ( n ) | n is finite. In 2004, Marcus and Tardos proved the Füredi–Hajnal conjecture, which together with the reduction introduced by Klazar in 2000 proves the Stanley–Wilf conjecture. We focus on the values of the Stanley–Wilf limit ( s P ) and the Füredi–Hajnal limit ( c P ). We improve the reduction and obtain s P ⩽ 2.88 c P 2 which decreases the general upper bound on s P from s P ⩽ const const O ( k log ( k ) ) to s P ⩽ const O ( k log ( k ) ) for any k × k permutation matrix P. In the opposite direction, we show c P = O ( s P 4.5 ) . For a lower bound, we present for each k a k × k permutation matrix satisfying c P = Ω ( k 2 ) .