Large monochromatic components and long monochromatic cycles in random hypergraphs

Bennett, Patrick J.; DeBiasio, Louis; Dudek, Andrzej; English, Sean

doi:10.1016/j.ejc.2018.10.001

Cited by 10 publications

(21 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In Section 2, we prove some general results giving bounds on H mc ( ) r for H an r-uniform hypergraph. We deduce an upper bound on S mc ( ) 3 for all Steiner triple systems S. We also prove a result which we will later use to deduce lower bounds on S mc ( ) 3 . We additionally give results on H mc ( ) r when H is a random r-uniform hypergraph.…”

Section: Overview Of the Papermentioning

confidence: 80%

“…In Section 4, we prove an absolute lower bound on S mc ( ) 3 for all Steiner triple systems S, namely, that…”

Section: Overview Of the Papermentioning

confidence: 99%

“…In Section 5, we prove that there exists an infinite family of Steiner triple systems  such that S o S mc ( ) = (2/3 + (1))| | 3 for all ∈ S . In Section 6, we discuss bounds on S mc ( ) 3 for Bose and Skolem triple systems S. Finally, in Section 7, we raise a number of Ramsey-type problems in the setting of Steiner triple systems. ).…”

Section: Overview Of the Papermentioning

confidence: 99%

“…In Theorem 1.2(i), Gyárfás proves an absolute lower bound on S mc ( ) 3 . Our first main result is a lower bound on S mc ( ) 3 in terms of α S ( ) * 3 .…”

mentioning

confidence: 94%

“…We then use Theorem 1.3 together with some recent methods developed by Kwan [19] to prove that for almost every ∈ S n  , we have S o n mc ( ) = (1 − (1)) 3 .…”

mentioning

confidence: 99%

See 4 more Smart Citations

Large monochromatic components in 3‐edge‐colored Steiner triple systems

DeBiasio

Tait

2020

J of Combinatorial Designs

Self Cite

View full text Add to dashboard Cite

It is known that in any r‐coloring of the edges of a complete r‐uniform hypergraph, there exists a spanning monochromatic component. Given a Steiner triple system on n vertices, what is the largest monochromatic component one can guarantee in an arbitrary 3‐coloring of the edges? Gyárfás proved that (2n+3)/3 is an absolute lower bound and that this lower bound is best possible for infinitely many n. On the other hand, we prove that for almost all Steiner triple systems the lower bound is actually (1−o(1))n. We obtain this result as a consequence of a more general theorem which shows that the lower bound depends on the size of a largest 3‐partite hole (ie, disjoint sets X1,X2,X3 with false|X1false|=false|X2false|=false|X3false| such that no edge intersects all of X1,X2,X3) in the Steiner triple system (Gyárfás previously observed that the upper bound depends on this parameter). Furthermore, we show that this lower bound is tight unless the structure of the Steiner triple system and the coloring of its edges are restricted in a certain way. We also suggest a variety of other Ramsey problems in the setting of Steiner triple systems.

show abstract

Section: Overview Of the Papermentioning

confidence: 80%

“…In Section 4, we prove an absolute lower bound on S mc ( ) 3 for all Steiner triple systems S, namely, that…”

Section: Overview Of the Papermentioning

confidence: 99%

Section: Overview Of the Papermentioning

confidence: 99%

“…In Theorem 1.2(i), Gyárfás proves an absolute lower bound on S mc ( ) 3 . Our first main result is a lower bound on S mc ( ) 3 in terms of α S ( ) * 3 .…”

mentioning

confidence: 94%

“…We then use Theorem 1.3 together with some recent methods developed by Kwan [19] to prove that for almost every ∈ S n  , we have S o n mc ( ) = (1 − (1)) 3 .…”

mentioning

confidence: 99%

See 3 more Smart Citations

Large monochromatic components in 3‐edge‐colored Steiner triple systems

DeBiasio

Tait

2020

J of Combinatorial Designs

Self Cite

View full text Add to dashboard Cite

show abstract

Covering random graphs with monochromatic trees

Domagoj

Bucić

2022

Random Struct Algorithms

View full text Add to dashboard Cite

Given an r-edge-colored complete graph K n , how many monochromatic connected components does one need in order to cover its vertex set? This natural question is a well-known essentially equivalent formulation of the classical Ryser's conjecture which, despite a lot of attention over the last 50 years, still remains open. A number of recent papers consider a sparse random analogue of this question, asking for the minimum number of monochromatic components needed to cover the vertex set of an r-edge-colored random graph (n, p). Recently, Bucić, Korándi, and Sudakov established a connection between this problem and a certain Helly-type local to global question for hypergraphs raised about 30 years ago by Erdős, Hajnal, and Tuza. We identify a modified version of the hypergraph problem which controls the answer to the problem of covering random graphs with monochromatic components more precisely. To showcase the power of our approach, we essentially resolve the 3-color case by showing that (log n∕n) 1∕4 is a threshold at which point three monochromatic components are needed to cover all vertices of a 3-edge-colored random graph, answering a question posed by Kohayakawa, Mendonça, Mota, and Schülke. Our approach also allows us to determine the answer in the general r-edge colored instance of the problem, up to lower order terms, around the point when it first becomes bounded, answering a question of Bucić, Korándi, and Sudakov.

show abstract