格路径的奇镇和偶镇定理
Oddtown and eventown theorems for lattice paths
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中文总结 AI 辅助
研究东北格路径，通过公共边定义交集，证明了公共边为偶数的路径族大小至多\(2^n\)，给出公共边为奇数时路径族最大大小的界及构造，还构造了至少\(C_n\)个极值偶交集族并猜想其为所有极值族。
中文摘要 AI 辅助
对于东北格路径（简称为格路径），我们通过公共边来定义交集。我们证明了从\((0,0)\)到\((n,n)\)的一族路径中，任意两条不同路径的公共边数量为偶数时，其大小至多为\(2^n\)，且该界可达。若\(M_{\mathrm{odd}}(n)\)表示任意两条不同路径的公共边数量为奇数的族的最大大小，我们证明\(M_{\mathrm{odd}}(n)\leq n(n - 1)+1\)，并构造出表明\(M_{\mathrm{odd}}(n)=\Theta(n^2)\)的族。最后，我们构造了至少\(C_n\)个不同的极值偶交集族，其中\(C_n\)是第\(n\)个卡特兰数，并猜想这些就是所有的极值族。
英文摘要
For North-East lattice paths (which we simply call lattice paths), we define intersection in terms of common edges. We prove that a family of paths from $(0,0)$ to $(n,n)$ in which every two distinct paths have an even number of common edges has size at most $2^n$, and that this bound is attained. If $M_{\mathrm{odd}}(n)$ denotes the maximum size of a family in which every two distinct paths have an odd number of common edges, then we prove \[ M_{\mathrm{odd}}(n)\le n(n-1)+1 \] and construct families showing that $M_{\mathrm{odd}}(n)=Θ(n^2)$. Finally, we construct at least $C_n$ distinct extremal even-intersecting families, where $C_n$ is the $n$th Catalan number, and conjecture that these are all the extremal families.
