Degree Power Sums in Extremal Set Systems

Abstract

For a family $\mathcal{F}\subseteq \left(\genfrac{}{}{0ex}{}{[n]}{k}\right)$ and $R\in \left(\genfrac{}{}{0ex}{}{[n]}{r}\right)$, let $d_{\mathcal{F}}(R)=|\{F\in \mathcal{F}:R\subseteq F\}|$ and ${\ell }_{r,p}(\mathcal{F})={\sum }_{R\in \left(\genfrac{}{}{0ex}{}{[n]}{r}\right)}{d}_{\mathcal{F}}(R{)}^p$; write $c{o}_p(\mathcal{F})={\ell }_{k-1,p}(\mathcal{F})$ for the codegree power sum. We introduce a method that uses convexity to extend sharp bounds for degree sums and sums of squared degrees to real powers, while retaining all equality cases. The method bounds $x^p$ by quadratic polynomials or by a continuous function that is linear on each of two intervals. These functions agree with $x^p$ at the degrees of the proposed extremal family, so the argument requires no bounds for sums of higher powers. For families with bounded matching number, we instead use a bound for ${\sum }_R\max \{d_{\mathcal{F}}(R)-s,0\}$ together with the degree sum. We give three exact applications. First, a full $t$-star maximizes $c{o}_p$ among $t$-intersecting families for every real $p\ge 2$ in the sharp range $n\ge (t+1)(k-t+1)$, with all equality cases determined. This extends the quadratic theorem of Wu and Zhang to real exponents and answers a problem of Zhou and Yuan throughout the sharp Erdős--Ko--Rado range. Second, among intersecting families with $n\ge 2k$, a full star maximizes ${\ell }_{r,p}$ for every $1\le r\le k-1$ and real $p\ge 2$, again with all equality cases determined. Third, if $\nu (\mathcal{F})\le s$ and $n\ge (2s+1)k-s$, then for every real $p\ge 1$, $c{o}_p(\mathcal{F})$ is uniquely maximized, up to permutation, by all $k$-sets meeting a fixed $s$-set. This removes the integrality restriction on $p$ and replaces previous cubic thresholds or assumptions that $n$ is sufficiently large with an explicit linear range valid for arbitrary $k$.