AI 中文总结
研究Fulek定义的0-1矩阵\(L_3\)，通过将矩阵元素分配给超图边、控制边多重性并归因于行上非交叉图的方法，证明了避免\(L_3\)的矩阵元素数量界，验证了关于线性轻模式的猜想。
AI 中文摘要
Fulek定义了0-1矩阵\[ L_3=\begin{pmatrix} 1&0&0&1&0\\ 0&0&0&0&1\\ 0&1&1&0&0 \end{pmatrix} \]并询问\(\text{ex}(n,L_3) = O(n)\)是否成立。我们证明了每个避免\(L_Three\)的\(r\times s\)0-1矩阵最多有\(27r + 2s\)个1元素。Fulek的一般下界构造有\(6n - 8\)个1元素，所以对于\(n\geq5\)，有\(6n - 8\leq \text{ex}(n,L_3)\leq29n\)。同样的论证适用于一个无限族。若\(Q_{a,b,k,\ell}\)是具有列字\(1^a3^k1^b2^\ell\)的轻三行矩阵，其中\(a,b,\ell\geq1\)且\(k\geq2\)，则\(\text{ex}(r,s,Q_{a,b,k,\ell}) \leq\bigl(5(k - 1)(4b + 1)+a + b+\ell - 1\bigr)r + 2s\)。这验证了Pettie和Tardos关于线性轻模式的一个猜想，该无限族包括之前未解决的权重为五的模式\(L_3\)。证明通过将矩阵元素分配给条形1-可见性超图的边，切割间隙以控制这些边的多重性，并将切割归因于行上的非交叉图。
英文摘要
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.
