Turán Numbers for 3-Uniform Linear Paths of Length 3

@article{Jackowska2016TurnNF,
  title={Tur{\'a}n Numbers for 3-Uniform Linear Paths of Length 3},
  author={Eliza Jackowska and Joanna Polcyn and Andrzej Rucinski},
  journal={Electron. J. Comb.},
  year={2016},
  volume={23},
  pages={2}
}
In this paper we confirm a conjecture of F\"uredi, Jiang, and Seiver, and determine an exact formula for the Tur\'an number $ex_3(n; P_3^3)$ of the 3-uniform linear path $P^3_3$ of length 3, valid for all $n$. It coincides with the analogous formula for the 3-uniform triangle $C^3_3$, obtained earlier by Frankl and F\"uredi for $n\ge 75$ and Cs\'ak\'any and Kahn for all $n$. In view of this coincidence, we also determine a `conditional' Tur\'an number, defined as the maximum number of edges in… 
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11 Citations
Highly Influential Citations
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Background Citations
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11 Citations

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The multipartite Ramsey number for the 3-path of length three
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On the Multicolor Ramsey Number for 3-Paths of Length Three
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On multicolor Ramsey numbers for loose k-paths of length three
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References

SHOWING 1-10 OF 19 REFERENCES
Refined Turán numbers and Ramsey numbers for the loose 3-uniform path of length three
Multicolor Ramsey Numbers and Restricted Turán Numbers for the Loose 3-Uniform Path of Length Three
TLDR
The largest number of edges in an $n$-vertex $P$-free 3-graph which is not a star is determined, which allows us to confirm the Tur\'an type formula, R(P;r) + 6 for r in 4,5,6,7.
Turán Numbers for Forests of Paths in Hypergraphs
TLDR
The results build on recent results of Furedi, Jiang, and Seiver, who determined the extremal numbers for individual paths, and provide more hypergraphs whose Turan numbers are exactly determined.
Exact solution of the hypergraph Turán problem for k-uniform linear paths
TLDR
The intensive use of the delta-system method is used to determine exk(n, Pℓ(k) exactly for all fixed ℓ ≥1, k≥4, and sufficiently large n, and describe the unique extremal family.
The 3-colored Ramsey number for a 3-uniform loose path of length 3
The values of hypergraph 2-color Ramsey numbers for loose cycles and paths have already been determined. The only known value for more than 2 colors is R(C 3 ; 3) = 8, where C 3 3 is a 3-uniform…
Exact solution of some Turán-type problems
Turán problems and shadows I: Paths and cycles
A PROBLEM ON INDEPENDENT r-TUPLES
then G(n ; l) contains k independent edges . It is easy to see that the above result is best possible since the complete graph of 2k-1 vertices and the graph of vertices x1, . . ., xk-1 ; Yl, • • •,…
Non-trivial intersecting families
Intersection Theorems for Systems of Sets
A version of Dirichlet's box argument asserts that given a positive integer a and any a2 +1 objects x0 , x1 , . . ., xa 2, there are always a+1 distinct indices v (0 < v < a 2) such that the…
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