A Sauer–Shelah–Perles lemma for lattices

Stijn Cambie, Bogdan Chronomaz, Zeev Dvir, Yuval Filmus and Shay Moran
Electronic Journal of Combinatorics

We extend the classical Sauer–Shelah–Perles lemma to lattices, using an argument which also appears in an unpublished monograph of Babai and Frankl on the linear algebra method in combinatorics.

The basic argument works only for lattices with nonvanishing Möbius function, but we are able to extend it to a slightly wider class of lattices.

The Sauer–Shelah–Perles lemma (in the strong form due to Pajor) fails for lattices with an induced copy of the lattice $0<1<2$. We conjecture that this is the only obstruction.


 author = {Stijn Cambie and Bogdan Chornomaz and Zeev Dvir and Yuval Filmus and Shay Moran},
 title = {A {S}auer--{S}helah--{P}erles Lemma for Lattices},
 journal = {Electronic Journal of Combinatorics},
 volume = {27},
 number = {4},
 pages = {P4.19},
 year = {2020}
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