combinatorics

Convex Transference for Degree Powers in Extremal Set Systems

arXiv:2607.28616

summary

The paper introduces a discrete two‑moment interpolation technique to bound sums of degree powers in intersecting families and shows that full star families uniquely maximize these sums for all real exponents p ≥ 2 within the Erdős–Ko–Rado range.

Abstract

For a family $F \subseteq (k [ n ])$ and $R \in (r [ n ])$ , let $d_{F} (R) = ∣ {F \in F : R \subseteq F} ∣$ and $ℓ_{r, p} (F) = \sum_{R \in (r [ n ])} d_{F} (R)^{p}$ ; at the codegree level, write $c o_{p} (F) = ℓ_{k - 1, p} (F)$ . We introduce a new convex-transference method for degree-power extremal problems and develop it into a reusable input--transfer--rigidity framework independent of any particular set-system problem. We give three exact applications. First, a full $t$ -star maximizes $c o_{p}$ among $t$ -intersecting families for every real $p \geq 2$ in the sharp range $n \geq (t + 1) (k - t + 1)$ , with all equality cases determined. This extends the Wu--Zhang quadratic theorem to real exponents and answers a problem of Zhou--Yuan throughout the sharp ErdÅs--Ko--Rado range. Second, if $n \geq 2 k$ , a full point-star maximizes $ℓ_{r, p}$ for every $1 \leq r \leq k - 1$ and real $p \geq 2$ , again with complete equality classification; thus the framework is not confined to codegrees. Third, if $I^½ (F) \leq s$ and $n \geq (2 s + 1) k - s$ , then for every real $p \geq 1$ , $c o_{p} (F)$ is uniquely maximized, up to isomorphism, by all $k$ -sets meeting a fixed $s$ -set. This removes the integrality restriction on $p$ and replaces previous cubic thresholds or nonexplicit sufficiently-large assumptions with an explicit linear range valid for arbitrary uniformity.

Topics & keywords

#intersecting families #degree powers #extremal set theory #codegree #Erdős–Ko–Rado theoremdegree power sumt‑starfull point‑starquadratic interpolationWu–Zhang bound