Degree Power Sums in Extremal Set Systems
For a family $\mathcal F\subseteq\binom{[n]}k$ and $R\in\binom{[n]}r$, let $d_{\mathcal F}(R)=|\{F\in\mathcal F:R\subseteq F\}|$ and $\ell_{r,p}(\mathcal F)=\sum_{R\in\binom{[n]}r}d_{\mathcal F}(R)^p$; write $co_p(\mathcal F)=\ell_{k-1,p}(\mathcal F)$ 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 $co_p$ among $t$-intersecting families for every real $p\geq2$ in the sharp range $n\geq(t+1)(k-t+1)$, 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 $n\geq2k$, a full star maximizes $\ell_{r,p}$ for every $1\leq r\leq k-1$ and real $p\geq2$, again with all equality cases determined. Third, if $ν(\mathcal F)\leq s$ and $n\geq(2s+1)k-s$, then for every real $p\geq1$, $co_p(\mathcal F)$ 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 $k$.