The Füredi-Hajnal Conjecture Implies the Stanley-Wilf Conjecture

Abstract

We show that the Stanley-Wilf enumerative conjecture on permutations follows easily from the Fiiredi-Hajnal extremal conjecture on 0-1 matrices. We apply the method, discovered by Alon and Friedgut, that derives an (almost) exponential bound on the number of some objects from a (almost) linear bound on their sizes. They proved by it a weaker form of the Stanley-Wilf conjecture. Using bipartite graphs, we give a simpler proof of their result. IIoKameM, 'ITO nmoTe3a CTaHJIH H BHJIljla 0 t{HCJIe IIepecTaHoBoK Bhl-TeKaeT IIpOCThlM 06pa30M H3 I'HIIOTe3hl H Xaii-HaJIa 0 0-1 MaTpHqax. IIpHMeHJIeM MeTO.n; BhlBo.n;a (IIO'lTH) aKCIIOHeH-qHaJILHOH oqeHKH 'IHCJIa H3 (IIO'lTH) oqeHKH HX BeJIH'IHH OTKPhlThlH AJIOHOM H i3THM OHH 3aJIH I'HIIOTe3Y CTaHJIH H BHJIljla B oCJIa6JIeHHOH ljIopMe. 'C I'paljloB IIOJIyt{HM 60JIee IIpocToe HX pe-3YJILTaTa. The Stanley-Wilf conjecture asserts that the number of n-pennutations not containing a given pennutation is exponential in n. Alon and Friedgut [IJ proved that it is true provided we have a linear upper bound on lengths of certain words over an ordered alphabet. They also proved a weaker version of it with an almost exponential upper bound. In the present note we want to inform the reader about this interesting development by reproving the latter result in a simpler way. We use bipartite graphs instead of words. We point out that in 1992 Fiiredi and Hajnal almost made an extremal conjecture on 0-1 matrices that now can easily be seen to imply the Stanley-Wilf conjecture. We prove that both extremal conjectures are logically equivalent. We use N to denote the set {I, 2, ... } and [nJ to denote the set {I, 2, ... , n}. The sets of all finite sequences (words) over N and [n] are denoted N* and [nJ*. If u E N*, lui is the length of u. A sequence v = b1b2 •.• bl E N* is k-sparse if bj = bi , j > i, implies j-i k. In other words, in each interval in v of length .. Supported by the grant GAUK 158/99.