非对称超图移除引理中丰度的紧致性
Compactness of abundance in asymmetric hypergraph removal lemmas
浏览论文内容
中文总结 AI 辅助
研究非对称超图移除引理中丰度的紧致性，证明丰富族含丰富成员及相关着色定理，给出非孤立核阶的显式界并应用于构造线性系统、获方程界、给测试器及证明判定问题不可解。
中文摘要 AI 辅助
固定整数\(r\geq2\)以及至少有一条边且无孤立顶点的有限简单\(r\) - 均匀超图\(F\)。若要删除至少\(\epsilon n^r\)条边才能破坏\(F\)的每一个副本，则称\(n\)个顶点的\(r\) - 图与\(F\) - 自由图的距离为\(\epsilon\)。若存在常数\(c,C\gt0\)，使得每个足够大的与\(F\) - 自由图距离为\(\epsilon\)的宿主图至少包含\(c\epsilon^C n^{v(H)}\)个带标记的\(H\)副本，则有限\(r\) - 图\(H\)是\(F\) - 丰富的。我们证明每个丰富的族都包含一个丰富的成员。对于包含\(F\)的边不相交且尊重部分的副本且每个顶点至少位于其中\(\epsilon n^{r - 1}\)个副本中的\(F\) - 部分宿主图，我们证明了类似的着色定理。\(r = 2\)的情况得出了着色和未着色图的紧致性定理，回答了Girão、Hurley、Illingworth和Michel的问题5.2，并证明了他们的猜想5.1。我们还得到了所选见证的非孤立核的阶的显式界\(\lfloor 2r(C + 1)\rfloor\)。此外，我们给出了几个应用。例如，我们从丰富的着色超图构造平移不变线性系统，为与有界长度循环相关的方程获得平方根界，基于一个固定图给出单边测试器，以及证明没有算法能判定由图灵机枚举的有限简单图族是否是\(K_3\) - 丰富的。
英文摘要
Fix an integer $r\ge 2$ and a finite simple $r$-uniform hypergraph $F$ with at least one edge and no isolated vertices. An $n$-vertex $r$-graph is $ε$-far from being $F$-free if at least $εn^r$ edges must be deleted to destroy every copy of $F$. A finite $r$-graph $H$ is $F$-abundant if there are constants $c,C>0$ such that every sufficiently large $ε$-far host contains at least $cε^C n^{v(H)}$ labelled copies of $H$. A family is $F$-abundant when one member has this lower bound in each host, although the member may depend on the host and on $ε$, while $c$ and $C$ are common to the family. We prove that every abundant family contains an abundant member. We prove the analogous coloured theorem for $F$-partite hosts containing edge-disjoint part-respecting copies of $F$ such that every vertex lies in at least $εn^{r-1}$ of them. The case $r=2$ yields the coloured and uncoloured graph compactness theorems, answers Question 5.2 of Girão, Hurley, Illingworth and Michel, and proves their Conjecture 5.1 [J. Lond. Math. Soc., 2024]. We also obtain an explicit bound $\lfloor 2r(C+1)\rfloor$ for the order of the non-isolated core of a selected witness. Moreover, we give several applications. For example, we construct translation-invariant linear systems from abundant coloured hypergraphs, obtain a square-root bound for an equation associated with a cycle of bounded length, give a one-sided tester based on one fixed graph when distance from the property gives a polynomial lower bound on distance from being $F$-free, and prove that no algorithm decides whether a family of finite simple graphs enumerated by a Turing machine is $K_3$-abundant.
