Completion and deficiency problems

Nenadov, Rajko; Sudakov, Benny; Wagner, Adam Zsolt

doi:10.48550/arxiv.1904.01394

Cited by 3 publications

(3 citation statements)

References 36 publications

(48 reference statements)

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“…We also mention the recent related work of Nenadov, Sudakov and Wagner [19] on embedding partial Steiner triple system to a small complete STS, and in general, embedding certain partial substructures to complete structures. In the spreading problem of linear 3-graphs, one may consider the triples of the hypergraph as collinearity prescription for triples of points, and under this condition the aim would be to embed the partial linear space to an affine of projective plane of small order.…”

Section: Further Results and Open Problemsmentioning

confidence: 99%

Spreading linear triple systems and expander triple systems

Blázsik¹,

Nagy²

2019

Preprint

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The existence of Steiner triple systems STS(n) of order n containing no nontrivial subsystem is well known for every admissible n. We generalize this result in two ways. First we define the expander property of 3-uniform hypergraphs and show the existence of Steiner triple systems which are almost perfect expanders.Next we define the strong and weak spreading property of linear hypergraphs, and determine the minimum size of a linear triple system with these properties, up to a small constant factor. A linear triple system on a vertex set V has the spreading, or respectively weakly spreading property if any sufficiently large subset V ⊂ V contains a pair of vertices with which a vertex of V \ V forms a triple of the system. Here the condition on V refers to |V | ≥ 4 or V is the support of more than one triples, respectively. This property is strongly connected to the connectivity of the structure the so-called influence maximization. We also discuss how the results are related to Erdős' conjecture on locally sparse STSs, subsquarefree Latin-squares and possible applications in finite geometry.

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Section: Further Results and Open Problemsmentioning

confidence: 99%

Spreading linear triple systems and expander triple systems

Blázsik¹,

Nagy²

2019

Preprint

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“…We also mention the recent related work of Nenadov, Sudakov and Wagner [65] on embedding partial Steiner triple system to a small complete STS, and in general, embedding certain partial substructures to complete structures. In the spreading problem of linear 3-graphs, one may consider the triples of the hypergraph as collinearity prescription for triples of points, and under this condition the aim would be to embed the partial linear space to an affine of projective plane of small order.…”

Section: Further Results and Open Problemsmentioning

confidence: 99%

Graphs and finite geometries

Blázsik¹

2019

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First and foremost, I am most grateful to my supervisor, Tamás Szőnyi, who helped me all the time by sharing his profound knowledge of finite geometry. He was so kind to spend countless hours with me providing guidance, suggesting new ideas which most of the time turned out to be essential ones. It has been a real privilege to be a student of Tamás Szőnyi.Besides Tamás, I am truly thankful to Zoltán Lóránt Nagy, who serves as a role model for me in many ways. On several occasions, he introduced me to new problems most of which I found extremely exciting, furthermore, his solid work ethic have been an example for me to follow. Zoltán also helped me tremendously by sharing his valuable insights about writing a paper.Without the help of György Kiss, I have not been able to take part in a wonderful research program at the Fields Institute in Toronto, Canada during my undergraduate years. I have learnt a lot there, not just the way of doing research in mathematics but to work in a team, communicating better and presenting in front of an illustrious audience. I will forever be indebted to him for this incredible opportunity. This experience, together with the fascinating lectures of the Finite Geometry Seminar got me interested in doing research in general.I would like to thank every current and past member of the MTA-ELTE Geometric and Algebraic Combinatorics Research Group for the excellent atmosphere and the stimulating discussions. It is a pleasure to work with such a friendly and helpful group of people.

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“…This paper concerns the problem of determining whether a given partial Steiner triple system has a small embedding of a specified order. Various aspects of this problem have been addressed in many papers (see [2,3,4,10,14] for example). In this paper we provide updates on two of these contributions, namely [4] and [2].…”

Section: Introductionmentioning

confidence: 99%

On determining when small embeddings of partial Steiner triple systems exist

Bryant

Gunasekara

Horsley

2020

J of Combinatorial Designs

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A partial Steiner triple system of order u is a pair (U, A) where U is a set of u elements and A is a set of triples of elements of U such that any two elements of U occur together in at most one triple. If each pair of elements occur together in exactly one triple it is a Steiner triple system. An embedding of a partial Steiner triple system (U, A) is a (complete) Steiner triple system (V, B) such that U ⊆ V and A ⊆ B. For a given partial Steiner triple system of order u it is known that an embedding of order v 2u + 1 exists whenever v satisfies the obvious necessary conditions. Determining whether "small" embeddings of order v < 2u + 1 exist is a more difficult task. Here we extend a result of Colbourn on the NP-completeness of these problems. We also exhibit a family of counterexamples to a conjecture concerning when small embeddings exist.

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Completion and deficiency problems

Cited by 3 publications

References 36 publications

Spreading linear triple systems and expander triple systems

Spreading linear triple systems and expander triple systems

Graphs and finite geometries

On determining when small embeddings of partial Steiner triple systems exist

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