Multivalued forbidden numbers of two-rowed configurations -- the missing cases

The present paper considers extremal combinatorics questions in the language of matrices. An s-matrix is a matrix with entries in \0,1,…, s-1\. An s-matrix is simple if it has no repeated columns. A matrix F is a configuration in a matrix A, denoted F A, if it is a row/column permutation of a submatrix of A. Avoid(m,s,F) is the set of m-rowed, simple s-matrices not containing a configuration of F and forb(m,s, F)=max\|A| A ∈ Avoid(m,s,F)\. Dillon and Sali initiated the systematic study of forb(m,s, F) for $2$-matrices F, and computed forb(m,s, F) for all 2-rowed F when $s>3$. In this paper we tackle the remaining cases when $s=3$. In particular, we determine the asymptotics of forb(m,3,p· K₂)-forb(m,3,p· I₂) for $p>3$, where K₂ is the 2× 4 simple $2$-matrix and I₂ is the 2× 2 identity matrix, as well as the exact values of forb(m,3,F) for many 2-rowed $2$-matrices F. 19 pages