Turán阈值以下色约束下的团超饱和问题
Clique supersaturation under a chromatic constraint below the Turán threshold
AI总结：
本文针对Turán阈值以下非p部图的场景，证明了满足边数条件的图中K_{p+1}副本数的精确团计数界，其构造达到该界，为Brouwer阈值提供了团计数类似物。
AI中文摘要：
极值图论的核心主题之一是超饱和问题，该问题研究在给定边条件下强制目标子图的最小副本数。该研究方向可追溯至Rademacher和Erdős针对三角形的研究，后续Lovász与Simonovits将其扩展至Turán阈值以上的团情形，Mubayi则进一步将该理论扩展至色临界图。在Turán阈值以下，非p部图场景中出现了密切相关的存在阈值现象：Brouwer的经典结果表明，当n≥2p+1时，每个n顶点非p部K_{p+1}-free图的边数至多为e(T_{n,p})−⌊n/p⌋+1。受该阈值启发，我们在非p部假设下研究Turán阈值以下的精确团计数问题。设p≥2、s≥1为固定整数，Y_{n,p,s}是通过在T_{n,p}的最大部分内添加一条边，并删除该新边的一个端点到最小部分的边中除s条外的所有边得到的图，其边数e(Y_{n,p,s})=e(T_{n,p})−⌊n/p⌋+s+1。我们证明，对于所有足够大的n，每个边数e(G)≥e(Y_{n,p,s})的n顶点非p部图G，包含的K_{p+1}副本数至少与Y_{n,p,s}中的数量相同，且该界是精确的，由构造Y_{n,p,s}达到。因此，我们的结果为非p部K_{p+1}-free图的Brouwer阈值提供了精确的团计数类似物。
英文摘要：
A central theme in extremal graph theory is the supersaturation problem, which investigates the minimum number of copies of a target subgraph forced by prescribed edge conditions. This line of research goes back to Rademacher and Erdős for triangles, and was later extended to cliques by Lovász and Simonovits in the regime above the Turán threshold. Mubayi further extended this theory to color-critical graphs. Below the Turán threshold, a closely related existence-threshold phenomenon arises in the non-$p$-partite setting: a classical result of Brouwer shows that, for $n\ge 2p+1$, every $n$-vertex non-$p$-partite $K_{p+1}$-free graph has at most $e(T_{n,p})-\lfloor n/p\rfloor+1$ edges. Motivated by this threshold, we investigate a sharp clique-counting problem below the Turán threshold under the non-$p$-partite assumption. Let $p\ge 2$ and $s\ge 1$ be fixed integers. Let $Y_{n,p,s}$ be the graph obtained from $T_{n,p}$ by adding an edge inside a largest part and deleting all but $s$ of the edges from one endpoint of this new edge to a smallest part. Then $e(Y_{n,p,s})=e(T_{n,p})-\lfloor n/p\rfloor+s+1$. We prove that, for all sufficiently large $n$, every $n$-vertex non-$p$-partite graph $G$ with $e(G)\ge e(Y_{n,p,s})$ contains at least as many copies of $K_{p+1}$ as $Y_{n,p,s}$ does. The bound is sharp, as it is attained by the construction $Y_{n,p,s}$. Thus our result provides the exact clique-counting analogue of Brouwer's threshold for non-$p$-partite $K_{p+1}$-free graphs.
