Small forbidden configurations V: Exact bounds for 4 × 2 cases

Abstract

The present paper continues the work begun by Anstee, Ferguson, Griggs, Kamoosi and Sali on small forbidden configurations. We define a matrix to besimpleif it is a (0, 1)-matrix with no repeated columns. LetFbe ak× (0, 1)-matrix (the forbidden configuration). AssumeAis anm×nsimple matrix which has no submatrix which is a row and column permutation ofF. We define forb (m, F) as the largestn, which would depend onmandF, so that such anAexists.DefineFabcdas the (a+b+c+d) × 2 matrix consisting ofarows of [11],brows of [10],crows of [01] anddrows of [00]. With the exception ofF2110, we compute forb (m; Fabcd) for all 4 × 2Fabcd. A number of cases follow easily from previous results and general observations. A number follow by clever inductions based on a single column such as forb (m; F1111) = 4m− 4 and forb (m; F1210) = forb (m; F1201) = forb (m; F0310) = (2m)+m+ 2 (proofs are different). A different idea proves forb (m; F0220) = (2m) + 2m− 1 with the forbidden configuration being related to a result of Kleitman. Our results suggest that determining forb (m; F2110) is heavily related to designs and we offer some constructions of matrices avoidingF2110using existing designs.