Degree Power Sums in Extremal Set Systems
Abstract
For a family ℱ⊆binom[n]k and R∈binom[n]r, let dℱ(R)=|{F∈ℱ:R⊆ F}| and ℓr,p(ℱ)=∑R∈binom[n]rdℱ(R)p; write cop(ℱ)=ℓk−1,p(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 xp by quadratic polynomials or by a continuous function that is linear on each of two intervals. These functions agree with xp 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 ∑Rmax{dℱ(R)−s,0} together with the degree sum. We give three exact applications. First, a full t-star maximizes cop among t-intersecting families for every real p≥2 in the sharp range n≥(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≥2k, a full star maximizes ℓr,p for every 1≤ r≤ k−1 and real p≥2, again with all equality cases determined. Third, if ν(ℱ)≤ s and n≥(2s+1)k−s, then for every real p≥1, cop(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.
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BibTeX
@misc{cao2026degree,
title = {{Degree Power Sums in Extremal Set Systems}},
author = {Mengyu Cao and Mei Lu and Haixiang Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.28616},
eprint = {2607.28616},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.28616},
}
RIS
TY - UNPB
TI - Degree Power Sums in Extremal Set Systems
AU - Mengyu Cao
AU - Mei Lu
AU - Haixiang Zhang
PY - 2026
DA - 2026/09/27
PB - arXiv
DO - 10.48550/arxiv.2607.28616
UR - https://arxiv.org/abs/2607.28616
AB - For a family ℱ⊆binom[n]k and R∈binom[n]r, let d_ℱ(R)=|{F∈ℱ:R⊆ F}| and ℓ_r,p(ℱ)=∑_R∈binom[n]rd_ℱ(R)^p; write co_p(ℱ)=ℓ_k−1,p(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 ∑_Rmax{d_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≥2 in the sharp range n≥(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≥2k, a full star maximizes ℓ_r,p for every 1≤ r≤ k−1 and real p≥2, again with all equality cases determined. Third, if ν(ℱ)≤ s and n≥(2s+1)k−s, then for every real p≥1, co_p(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.
ER -