2026/07/30 by Mengyu Cao, Mei Lu, Haixiang Zhang
Mathematics · #math.CO #msc:05C65 #msc:05D05
For a family F⊆\binom[n]k and R∈\binom[n]r, let dF(R)=|\F\inF:R⊆ F\| and ℓr,p(F)=∑_R∈\binom[n]rdF(R)p; at the codegree level we write cop(F)=ℓk-1,p(F). We develop a discrete two-moment interpolation principle that majorizes xp on the integer degree lattice by a quadratic interpolant and reduces every real exponent p≥2 to sharp bounds for the first two falling moments. We prove that a full t-star maximizes cop among t-intersecting families for every real p≥2 throughout the sharp classical range n≥(t+1)(k-t+1), and we determine all equality cases. Using Bey's size-sensitive quadratic inequality, we extend the same framework to every nontrivial degree level: if F is intersecting, n≥2k, and 1≤ r≤ k-1, then a full point-star maximizes ℓr,p(F) for every real p≥2, again with a complete equality classification. Thus the codegree theorem extends the sharp Wu--Zhang quadratic bound to every real p≥2, completes the quadratic boundary equality classification, contains the Brooks--Linz conjecture as its p=2 special case, and, for integer exponents p≥2, resolves the problem of Zhou--Yuan throughout the sharp Erdős--Ko--Rado range.