相交整数分拆：星形上界与各尺度反例

中文摘要 AI 辅助

两个整数分拆若至少含有t个共同部分（按重数计），则称它们t-相交。我们研究将n分成恰好k个正整数部分的最大t-相交族。典范t-星由至少含有t个1的分拆组成。应用Kupavskii的弱扩散定理，我们证明：对于每个固定的A>24及所有充分大的k，当n≥Ak^3时，该星形族是最大的，且该结论对1≤t<k一致成立。我们还给出了Borg猜想在每个相交尺度下的反例：对所有充分大的k及每个1≤d<k，可选取d/4≤t≤d及n=⌊tk^2/3⌋，使得某个t-相交族严格大于典范星形族。

英文摘要

Two integer partitions $t$-intersect if they have at least $t$ common parts, counted with multiplicity. We study the largest $t$-intersecting families of integer partitions of $n$ into exactly $k$ positive parts. The canonical $t$-star consists of the partitions containing at least $t$ ones. Applying Kupavskii's weak-spread theorem, we prove that this star is largest whenever $n\ge Ak^3$, for every fixed $A>24$ and all sufficiently large $k$, uniformly over $1\le t<k$. We also give counterexamples to Borg's conjecture at every intersection scale: for all sufficiently large $k$ and for every $1\le d<k$, one may choose $d/4\le t\le d$ and $n=\lfloor tk^2/3\rfloor$ so that a $t$-intersecting family is strictly larger than the canonical star.

发表机构

• Technische Universität Ilmenau（埃尔福特工业大学）
• Technische Hochschule Rosenheim（罗森海姆应用技术大学）

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