经典极值问题中的$(t,p)$-范数
The $(t,p)$-Norm in Classical Extremal Problems
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中文总结 AI 辅助
该研究确定了匹配数受限的$r$-均匀超图、$k$-相交族、不含$P_\ell^r$超图的最大$(t,p)$-范数及对应极值族，覆盖$p>1$和$0<p<1$等范围。
中文摘要 AI 辅助
给定整数$r>t\ge1$和实数$p>0$，$r$-均匀超图$\mathcal{H}$的$(t,p)$-范数$||\mathcal{H}||_{t,p}$是所有$t$-子集$T\subseteq V(\mathcal{H})$的度数$d_{\mathcal{H}}(T)$的$p$次幂之和。当$t=r-1$时，该范数即为余度数$p$-范数。对于所有足够大的$n$，我们得到以下结果：前两个结果适用于凸范围$p>1$和凹范围$0<p<1$两种情况。其一，对于匹配数不超过$s$的$r$-均匀超图，我们确定了其最大$(t,p)$-范数；其二，对于$k$-相交族，我们建立了关于$(t,p)$-范数的Erdős–Ko–Rado型定理；其三，对于不含$P_\ell^r$的超图，我们对所有$1\le t\le r-1$和$p>1$的情况确定了其最大$(t,p)$-范数。在这三种情形中，我们还刻画了所有极值族。
英文摘要
Given integers $r>t\ge1$ and a real number $p>0$, the $(t,p)$-norm $||\mathcal{H}||_{t,p}$ of an $r$-graph $\mathcal{H}$ is the sum of the $p$-th powers of the degrees $d_{\mathcal{H}}(T)$ over all $t$-subsets $T\subseteq V(\mathcal{H})$. When $t=r-1$, this is the codegree $p$-norm. For all sufficiently large $n$, we obtain the following results. The first two apply in both the convex range $p>1$ and the concave range $0<p<1$. First, for $r$-graphs with matching number at most $s$, we determine the maximum $(t,p)$-norm. Second, for $k$-intersecting families, we establish an Erdős--Ko--Rado-type theorem for the $(t,p)$-norm. Third, for $P_\ell^r$-free hypergraphs, we determine the maximum $(t,p)$-norm for every $1\le t\le r-1$ and $p>1$. In each of the three settings, we also characterize all extremal families.
