Tight general bounds for the extremal numbers of 0-1 matrices

Abstract

A zero-one matrix $M$ is said to contain another zero-one matrix $A$ if we can delete some rows and columns of $M$ and replace some $1$-entries with $0$-entries such that the resulting matrix is $A$. The extremal number of $A$, denoted $\operatorname{ex}(n,A)$, is the maximum number of $1$-entries that an $n\times n$ zero-one matrix can have without containing $A$. The systematic study of this function for various patterns $A$ goes back to the work of Füredi and Hajnal from 1992, and the field has many connections to other areas of mathematics and theoretical computer science. The problem has been particularly extensively studied for so-called acyclic matrices, but very little is known about the general case (that is, the case where $A$ is not necessarily acyclic). We prove the first asymptotically tight general result by showing that if $A$ has at most $t$ $1$-entries in every row, then $\operatorname{ex}(n,A)\leq n^{2-1/t+o(1)}$. This verifies a conjecture of Methuku and Tomon. Our result also provides the first tight general bound for the extremal number of vertex-ordered graphs with interval chromatic number $2$, generalizing a celebrated result of Füredi, and Alon, Krivelevich and Sudakov about the (unordered) extremal number of bipartite graphs with maximum degree $t$ in one of the vertex classes.

Figures (2)

• Figure 1: Step $j$ is 'below-shrinking' for $c_1c_2$ because we have a row $i$ 'below' $M_j$ that contains a $1$-entry in both of the columns $c_1$ and $c_2$.
• Figure 2: The figure shows how to embed $A = 101011110$ into $M$ using 3 below-shrinking steps: we use step $j_1$ for the column pair $y_1 y_2$ to embed the third row of $A$, step $j_2$ for the column pair $y_2y_3$ to embed the second row of $A$, and step $j_3$ for the column pair $y_1 y_3$ to embed the first row of $A$.

Theorems & Definitions (9)

• Theorem 1.1: Füredi furedi1991turan, Alon--Krivelevich--Sudakov alon2003turan
• Theorem 1.2: Methuku--Tomon methuku2022bipartite
• Conjecture 1.3: Methuku--Tomon methuku2022bipartite
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6: Pach--Tardos pach2006forbidden
• Theorem 1.7
• Lemma 2.1
• Theorem 2.2