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Improved Upper Bound on the Linear Tur\'an Number of the Crown
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abstract
A linear $3$-graph is a set of vertices along with a set of edges, which are three element subsets of the vertices, such that any two edges intersect in at most one vertex. The crown, $C$, is a specific $3$-graph consisting of three pairwise disjoint edges, called jewels, along with a fourth edge intersecting all three jewels. For a linear $3$-graph, $F$, the linear Tur\'an number, $ex(n,F)$, is the maximum number of edges in any linear $3$-graph that does not contain $F$ as a subgraph. Currently, the best known bounds on the linear Tur\'an number of the crown are \[ 6 \Big \lfloor \frac{n-3}{4}\Big \rfloor \leq ex(n, C) \leq 2n. \] In this paper, the upper bound is improved to $ex(n,C) < \frac{5n}{3}$.
Forward citations
Cited by 3 Pith papers
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Robust Repulsion for Growing Crowns in Linear Hypergraphs
Sharp edges in linear crown-free hypergraphs are exactly the common lines of k-1 glued projective planes, and this rigidity yields a strict improvement to the extremal edge bound.
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Linear Tur\'an Numbers of Uniform Hypertrees
For several r-uniform linear hypertrees with four edges, the maximum number of edges in a linear r-uniform hypergraph avoiding them is determined; the 4-uniform 4-edge path case is settled exactly.
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Bounds on Linear Tur\'{a}n Number for Trees
The paper proves exact and near-exact linear Turán bounds for several small r-uniform trees, including a lower bound for the 4-edge path, but one upper-bound proof contains an unjustified assumption.
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