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Improved Upper Bound on the Linear Tur\'an Number of the Crown

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arxiv 2109.02729 v1 pith:FIHAO5KG submitted 2021-09-06 math.CO

Willem Fletcher This is my paper
classification math.CO
keywords linearedgesgraphnumbercrownthreealongbound
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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 $ex(n,C) < \frac{5n}{3}$.

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Cited by 4 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Linear Tur\'an Numbers of Uniform Hypertrees

    math.CO 2026-07 conditional novelty 6.0 of 10

    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.

  2. An Upper Bound on the Linear Tur\'{a}n Number of $k$-Crowns

    math.CO 2026-04 unverdicted novelty 6.0 of 10

    An upper bound is established for the linear Turán number ex_r^lin(n, C_{1,k}^r) of k-crowns in linear r-graphs.

  3. Bounds on Linear Tur\'{a}n Number for Trees

    math.CO 2026-01 unverdicted novelty 6.0 of 10

    Linear Turán number ex_r^lin(n,T_k^r) is at least n(k-1)/r for any r-uniform tree with k edges; exact upper bound (r+1)n/r for B_4^r with characterization, (2r-1)n/r for E_4^r, and matching lower construction for P_4^r.

  4. Bounds on Linear Tur\'{a}n Number for Trees

    math.CO 2026-01 conditional novelty 6.0 of 10

    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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