# 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}
}
• Published 11 June 2015
• Mathematics
• Electron. J. Comb.
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…
11 Citations

## Figures from this paper

Anti-Ramsey Numbers of Paths and Cycles in Hypergraphs
• Mathematics
SIAM J. Discret. Math.
• 2020
The anti-Ramsey numbers of linear paths and loose paths in hypergraphs for sufficiently large $n$ are determined, and bounds for the anti- Ramsey numbers of Berge paths are given.
On multicolor Ramsey numbers of triple system paths of length 3
• Mathematics
• 2019
Let $\mathcal{H}$ be a 3-uniform hypergraph. The multicolor Ramsey number $r_k(\mathcal{H})$ is the smallest integer $n$ such that every coloring of $\binom{[n]}{3}$ with $k$ colors has a
One More Turán Number and Ramsey Number for the Loose 3-Uniform Path of Length Three
This paper refined this analysis further and compute the fifth order Turán number for P, for all n, and confirms the formula R(P; 10) = 16.
Anti-Ramsey number of paths in hypergraphs
• Mathematics
• 2019
The anti-Ramsey number of a hypergraph $\mathcal{H}$ is the smallest integer $c$ such that in any coloring of the edges of the $s$-uniform complete hypergraph on $n$ vertices with exactly $c$ colors,
Monochromatic loose paths in multicolored k-uniform cliques
• Mathematics
Discret. Math. Theor. Comput. Sci.
• 2019
There is an algorithm such that for every $r$-edge-coloring of the edges of the complete $k$-uniform hypergraph, it finds a monochromatic copy of P_\ell^{(k)}$in time at most$cn^k$. Turán and Ramsey numbers for 3‐uniform minimal paths of length 4 • Mathematics J. Graph Theory • 2021 The second and third order Turan numbers are established and used to compute the corresponding Ramsey numbers for up to four colors. On the Multicolor Ramsey Number for 3-Paths of Length Three • Mathematics Electron. J. Comb. • 2017 We show that if we color the hyperedges of the complete$3$-uniform complete graph on$2n+\sqrt{18n+1}+2$vertices with$n$colors, then one of the color classes contains a loose path of length three. ## References SHOWING 1-10 OF 19 REFERENCES Multicolor Ramsey Numbers and Restricted Turán Numbers for the Loose 3-Uniform Path of Length Three • Mathematics Electron. J. Comb. • 2017 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
• Mathematics
SIAM J. Discret. Math.
• 2014
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
• Mathematics
Comb.
• 2014
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
• Mathematics
J. Comb. Theory, Ser. A
• 1987
Turán problems and shadows I: Paths and cycles
• Mathematics
J. Comb. Theory, Ser. A
• 2015
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
• Mathematics
J. Comb. Theory, Ser. A
• 1986
Intersection Theorems for Systems of Sets
• Mathematics
• 1960
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