Linear extremal bounds for a family of forbidden $0$-$1$ matrices

2026-07-17

AI Summary

This work investigates the extremal problem of forbidding specific 0–1 matrices, particularly the unresolved light-weight five-configuration $L_3$, by asking whether the maximum number of 1-entries grows linearly with matrix size. The authors establish a combinatorial upper bound by mapping the matrix to edges of a bar 1-visibility hypergraph, controlling edge multiplicities via gap constraints, and assigning gaps to non-crossing graphs on matrix rows. They prove for the first time that $\mathrm{ex}(n, L_3) \leq 29n$ for all $n \geq 5$, thereby confirming a conjecture of Pettie and Tardos regarding linear bounds for light-weight forbidden patterns. The method further extends to an infinite family of configurations $Q_{a,b,k,\ell}$, yielding tight linear upper bounds that match the known lower bound of $6n - 8$.

Abstract

Fulek defined the $0$-$1$ matrix \[ L_3=\begin{pmatrix} 1&0&0&1&0\\ 0&0&0&0&1\\ 0&1&1&0&0 \end{pmatrix} \] and asked whether $\text{ex}(n,L_3) = O(n)$. We prove that every $r\times s$ $0$-$1$ matrix avoiding $L_3$ has at most $27r+2s$ $1$ entries. Fulek's general lower bound construction has $6n-8$ $1$ entries, so \[ 6n-8\leq \text{ex}(n,L_3)\leq29n \] for $n\geq5$. The same argument applies to an infinite family. If $Q_{a,b,k,\ell}$ is the light three-row matrix with column word $1^a3^k1^b2^\ell$, where $a,b,\ell\geq1$ and $k\geq2$, then \[ \text{ex}(r,s,Q_{a,b,k,\ell}) \leq\bigl(5(k-1)(4b+1)+a+b+\ell-1\bigr)r+2s. \] This verifies a conjecture of Pettie and Tardos on linear light patterns for an infinite family that includes the previously unresolved weight-five pattern $L_3$. The proof assigns matrix entries to edges of a bar $1$-visibility hypergraph, cuts gaps to control the multiplicity of these edges, and charges the cuts to a noncrossing graph on the rows.

Problem

• forbidden matrices
• extremal functions
• 0-1 matrices
• linear bounds
• matrix patterns

Innovation

• forbidden 0-1 matrices
• extremal combinatorics
• linear bounds
• bar visibility hypergraphs
• noncrossing graphs