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

Fulek defined the $0$-$1$ matrix $$L_3=\left(\begin{array}{ccccc}1& 0& 0& 1& 0\\ 0& 0& 0& 0& 1\\ 0& 1& 1& 0& 0\end{array}\right)$$ 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 $$6n-8\le \text{ex}(n,L_3)\le 29n$$ for $n\ge 5$. The same argument applies to an infinite family. If $Q_{a,b,k,\mathrm{\ell }}$ is the light three-row matrix with column word $1^a{3}^k{1}^b{2}^{\mathrm{\ell }}$, where $a,b,\mathrm{\ell }\ge 1$ and $k\ge 2$, then $$\text{ex}(r,s,Q_{a,b,k,\mathrm{\ell }})\le (5(k-1)(4b+1)+a+b+\mathrm{\ell }-1)r+2s\mathrm{.}$$ 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.