Linear bound on extremal functions of some forbidden patterns in 0–1 matrices

In this note by saying that a 0–1 matrix A avoids a pattern P given as a 0–1 matrix we mean that no submatrix of A either equals P or can be transformed into P by replacing some 1 entries with 0 entries. We present a new method for estimating the maximal number of the 1 entries in a matrix that avoids a certain pattern. Applying this method we give a linear bound on the maximal number ex ( n , L 1 ) of the 1 entries in an n by n matrix avoiding pattern L 1 and thereby we answer the question that was asked by Gábor Tardos. Furthermore, we use our approach on patterns related to L 1 .