Robust Repulsion for Growing Crowns in Linear Hypergraphs

Abstract

Put $q=r-1$, $t=k-1$, and $D=tq+1$. For an edge $e$ of a linear $C_{1,k}^r$-free $r$-uniform hypergraph, define \[ δ_H(e)=\sum_{v\in e}\frac{1}{d_H(v)}-\frac{r}{D}. \] The defect satisfies ${\delta }_H(e)\ge 0$. At equality, every vertex of $e$ has degree $D$, and the petal trace at $e$ is a disjoint union of $t$ affine planes of order $q$. Equivalently, restoring the base line gives $t$ projective planes of order $q$ with common line $e$. For growing crowns, the equality structure is stable in the following sense. If $q_j\to \infty$, $2\le t_j\le q_j$, and $e_j$ is an edge of a finite linear $C_{1,t_j+1}^{q_j+1}$-free hypergraph satisfying \[ \frac{t_j^2}{q_j}\to0, \qquad t_j^3δ_{H_j}(e_j)\to0, \] then, for every fixed $0<\theta <1$, \[ \frac{ |\{f\ne e_j:f\cap e_j\ne\varnothing,\ δ_{H_j}(f)\geθ/t_j^2\}| }{(q_j+1)(t_jq_j)} \to1. \] Thus an edge close to equality is adjacent almost entirely to edges with defect of order at least $t_j^{-2}$. A uniform form gives absolute constants $Q_0,{\epsilon }_0,c_0>0$ such that, whenever $q\ge Q_0{t}^2$ and ${\delta }_H(e)<{\epsilon }_0/t^3$, at least $\frac{1}{2}(q+1)tq$ neighbors of $e$ have defect at least $1/(100t^2)$. Consequently, \[ c^{\mathrm{lin}}_{q+1,t+1}\le t-\frac{c_0}{t}, \] where $c_{r,k}^{\mathrm{lin}}$ denotes the asymptotic linear Turán coefficient for $C_{1,k}^r$.

16 pages