The maximum number of unit distances in a convex n-gon
Journal of Combinatorial Theory Series A
An extremal problem on sparse 0-1 matrices
SIAM Journal on Discrete Mathematics
Davenport-Schnizel theory of matrices
Discrete Mathematics
Excluded permutation matrices and the Stanley-Wilf conjecture
Journal of Combinatorial Theory Series A
On 0-1 matrices and small excluded submatrices
Journal of Combinatorial Theory Series A
Note: Extremal functions of forbidden double permutation matrices
Journal of Combinatorial Theory Series A
On nonlinear forbidden 0-1 matrices: a refutation of a Füredi-Hajnal conjecture
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Applications of forbidden 0-1 matrices to search tree and path compression-based data structures
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Generalized Davenport-Schinzel sequences and their 0-1 matrix counterparts
Journal of Combinatorial Theory Series A
On the structure and composition of forbidden sequences, with geometric applications
Proceedings of the twenty-seventh annual symposium on Computational geometry
Origins of Nonlinearity in Davenport-Schinzel Sequences
SIAM Journal on Discrete Mathematics
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In this paper we study the extremal problem of finding how many 1 entries an n by n 0-1 matrix can have if it does not contain certain forbidden patterns as submatrices. We call the number of 1 entries of a 0-1 matrix its weight. The extremal function of a pattern is the maximum weight of an n by n 0-1 matrix that does not contain this pattern as a submatrix. We call a pattern (a 0-1 matrix) linear if its extremal function is O(n). Our main results are modest steps towards the elusive goal of characterizing linear patterns. We find novel ways to generate new linear patterns from known ones and use this to prove the linearity of some patterns. We also find the first minimal non-linear pattern of weight above 4. We also propose an infinite sequence of patterns that we conjecture to be minimal non-linear but have @W(nlogn) as their extremal function. We prove a weaker statement only, namely that there are infinitely many minimal not quasi-linear patterns among the submatrices of these matrices. For the definition of these terms see below.