# On triple systems with independent neighbourhoods

Z. Füredi, Oleg Pikhurko, M. Simonovits

Research output: Article

24 Citations (Scopus)

Let ℋ be a 3-uniform hypergraph on an $n$-element vertex set $V$. The neighbourhood of $a,b\in V$ is $N(ab):= {x: abx ∈ (ℋ)}$. Such a 3-graph has independent neighbourhoods if no $N(ab)$ contains an edge of ℋ. This is equivalent to ℋ not containing a copy of double struck F sign3,2 :={abx, aby, abz, xyz}. In this paper we prove an analogue of the Andrásfai-Erdös-Sós theorem for triangle-free graphs with minimum degree exceeding $2n/5$. It is shown that any double struck F sign-free 3-graph with minimum degree exceeding $(4/9-1/125){n} {2} is bipartite, (for n > n0), i.e., the vertices of ℋ can be split into two parts so that every triple meets both parts. This is, in fact, a Turán-type result. It solves a problem of Erdös and T.Sós, and answers a question of Mubayi and Rödl that ex(n,double struck F sign3,2)=max(α}(n-α)\left({α 2}. Here the right-hand side is 4/9{n}{3}+O(n2)$. Moreover e(ℋ)=ex(n,double struck F sign3,2) is possible only if V(ℋ)$can be partitioned into two sets$A$and$B$so that each triple of ℋ intersects$A$in exactly two vertices and$B$in one. Original language English 795-813 19 Combinatorics Probability and Computing 14 5-6 https://doi.org/10.1017/S0963548305006905 Published - nov. 2005 ### Fingerprint Triple System Minimum Degree Triangle-free Graph Uniform Hypergraph Graph in graph theory Intersect Analogue Vertex of a graph Theorem ### ASJC Scopus subject areas • Theoretical Computer Science • Computational Theory and Mathematics • Mathematics(all) • Discrete Mathematics and Combinatorics • Statistics and Probability ### Cite this In: Combinatorics Probability and Computing, Vol. 14, No. 5-6, 11.2005, p. 795-813. Research output: Article @article{71e6beff013e4ef79d40681998aac742, title = "On triple systems with independent neighbourhoods", abstract = "Let ℋ be a 3-uniform hypergraph on an$n$-element vertex set$V$. The neighbourhood of$a,b\in V$is$N(ab):= {x: abx ∈ (ℋ)} $. Such a 3-graph has independent neighbourhoods if no$N(ab)$contains an edge of ℋ. This is equivalent to ℋ not containing a copy of double struck F sign3,2 :={abx, aby, abz, xyz}. In this paper we prove an analogue of the Andr{\'a}sfai-Erd{\"o}s-S{\'o}s theorem for triangle-free graphs with minimum degree exceeding$2n/5$. It is shown that any double struck F sign-free 3-graph with minimum degree exceeding$(4/9-1/125){n} {2} is bipartite, (for n > n0), i.e., the vertices of ℋ can be split into two parts so that every triple meets both parts. This is, in fact, a Tur{\'a}n-type result. It solves a problem of Erd{\"o}s and T.S{\'o}s, and answers a question of Mubayi and R{\"o}dl that ex(n,double struck F sign3,2)=max(α}(n-α)\left({α 2}. Here the right-hand side is 4/9{n}{3}+O(n2)$. Moreover e(ℋ)=ex(n,double struck F sign3,2) is possible only if V(ℋ)$ can be partitioned into two sets $A$ and $B$ so that each triple of ℋ intersects $A$ in exactly two vertices and $B$ in one.",
author = "Z. F{\"u}redi and Oleg Pikhurko and M. Simonovits",
year = "2005",
month = "11",
doi = "10.1017/S0963548305006905",
language = "English",
volume = "14",
pages = "795--813",
journal = "Combinatorics Probability and Computing",
issn = "0963-5483",
publisher = "Cambridge University Press",
number = "5-6",

}

TY - JOUR

T1 - On triple systems with independent neighbourhoods

AU - Füredi, Z.

AU - Pikhurko, Oleg

AU - Simonovits, M.

PY - 2005/11

Y1 - 2005/11

N2 - Let ℋ be a 3-uniform hypergraph on an $n$-element vertex set $V$. The neighbourhood of $a,b\in V$ is $N(ab):= {x: abx ∈ (ℋ)}$. Such a 3-graph has independent neighbourhoods if no $N(ab)$ contains an edge of ℋ. This is equivalent to ℋ not containing a copy of double struck F sign3,2 :={abx, aby, abz, xyz}. In this paper we prove an analogue of the Andrásfai-Erdös-Sós theorem for triangle-free graphs with minimum degree exceeding $2n/5$. It is shown that any double struck F sign-free 3-graph with minimum degree exceeding $(4/9-1/125){n} {2} is bipartite, (for n > n0), i.e., the vertices of ℋ can be split into two parts so that every triple meets both parts. This is, in fact, a Turán-type result. It solves a problem of Erdös and T.Sós, and answers a question of Mubayi and Rödl that ex(n,double struck F sign3,2)=max(α}(n-α)\left({α 2}. Here the right-hand side is 4/9{n}{3}+O(n2)$. Moreover e(ℋ)=ex(n,double struck F sign3,2) is possible only if V(ℋ)$can be partitioned into two sets$A$and$B$so that each triple of ℋ intersects$A$in exactly two vertices and$B$in one. AB - Let ℋ be a 3-uniform hypergraph on an$n$-element vertex set$V$. The neighbourhood of$a,b\in V$is$N(ab):= {x: abx ∈ (ℋ)} $. Such a 3-graph has independent neighbourhoods if no$N(ab)$contains an edge of ℋ. This is equivalent to ℋ not containing a copy of double struck F sign3,2 :={abx, aby, abz, xyz}. In this paper we prove an analogue of the Andrásfai-Erdös-Sós theorem for triangle-free graphs with minimum degree exceeding$2n/5$. It is shown that any double struck F sign-free 3-graph with minimum degree exceeding$(4/9-1/125){n} {2} is bipartite, (for n > n0), i.e., the vertices of ℋ can be split into two parts so that every triple meets both parts. This is, in fact, a Turán-type result. It solves a problem of Erdös and T.Sós, and answers a question of Mubayi and Rödl that ex(n,double struck F sign3,2)=max(α}(n-α)\left({α 2}. Here the right-hand side is 4/9{n}{3}+O(n2)$. Moreover e(ℋ)=ex(n,double struck F sign3,2) is possible only if V(ℋ)$ can be partitioned into two sets $A$ and $B$ so that each triple of ℋ intersects $A$ in exactly two vertices and $B$ in one.

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U2 - 10.1017/S0963548305006905

DO - 10.1017/S0963548305006905

M3 - Article

AN - SCOPUS:26644460016

VL - 14

SP - 795

EP - 813

JO - Combinatorics Probability and Computing

JF - Combinatorics Probability and Computing

SN - 0963-5483

IS - 5-6

ER -