The number of K m,m -free graphs

Balogh, József; Samotij, Wojciech

doi:10.1007/s00493-011-2610-y

Cited by 45 publications

(83 citation statements)

References 18 publications

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“…This is implied by the finding that the slope of the red‐spiral fraction versus local density relation declines smoothly and does not entirely flatten‐off at low densities. For example, such a conclusion is reached by Balogh et al (2004) due to a similar behaviour of the star‐forming fraction versus local density. However, when considering groupocentric distance as the environmental indicator, in Fig.…”

Section: Comparing Morphology and Colourmentioning

confidence: 68%

“…They also find an indication that star formation is enhanced in intermediate‐density environments at z ∼ 0.6 (see also Bamford et al 2005), in contrast to the independence (e.g. Balogh et al 2004) or suppression (e.g. Koopmann & Kenney 2004) of star formation with respect to increasing local density seen nowadays.…”

Section: Comparing Morphology and Colourmentioning

confidence: 94%

“…As mentioned above, environmental trends in galaxy colours and star formation rates have received much attention recently (Lewis et al 2002; Pimbblet et al 2002; Gómez et al 2003; Baldry et al 2004, 2006; Balogh et al 2004; Kauffmann et al 2004; Blanton et al 2005; Weinmann et al 2006; Blanton & Berlind 2007). The general result is that the fraction of blue (star‐forming) galaxies decreases with local density, in favour of red (passive) objects.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Galaxy Zoo: the dependence of morphology and colour on environment

Bamford

Nichol

Baldry

et al. 2009

Monthly Notices of the Royal Astronomical Society

555

714

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Does the environment of a galaxy directly influence the quenching history of a galaxy? Here we investigate the detailed morphological structures and star formation histories of a sample of SDSS group galaxies with both classifications from Galaxy Zoo 2 and NUV detections in GALEX. We use the optical and NUV colours to infer the quenching time and rate describing a simple exponentially declining SFH for each galaxy, along with a control sample of field galaxies. We find that the time since quenching and the rate of quenching do not correlate with the relative velocity of a satellite but are correlated with the group potential. This quenching occurs within an average quenching timescale of ∼ 2.5 Gyr from star forming to complete quiescence, during an average infall time (from ∼ 10R 200 to 0.01R 200) of ∼ 2.6 Gyr. Our results suggest that the environment does play a direct role in galaxy quenching through quenching mechanisms which are correlated with the group potential, such as harassment, interactions or starvation. Environmental quenching mechanisms which are correlated with satellite velocity, such as ram pressure stripping, are not the main cause of quenching in the group environment. We find that no single mechanism dominates over another, except in the most extreme environments or masses. Instead an interplay of mergers , mass & morphological quenching and environment driven quenching mechanisms dependent on the group potential drive galaxy evolution in groups.

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Section: Comparing Morphology and Colourmentioning

confidence: 68%

Section: Comparing Morphology and Colourmentioning

confidence: 94%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Galaxy Zoo: the dependence of morphology and colour on environment

Bamford

Nichol

Baldry

et al. 2009

Monthly Notices of the Royal Astronomical Society

555

714

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show abstract

“…Recently, Balogh and Samotij [9,10] refined the ideas of Kleitman and Winston and obtained a bound on the number of independent sets in uniform hypergraphs satisfying a similar local density condition. Even more recently, Alon, Balogh, Morris and Samotij [1] used similar ideas to bound the number of independent sets in 'almost linear' 3-uniform hypergraphs satisfying a more general density condition termed (α, B)-stability, see Definition 6.1.…”

Section: Proof Of the Main Theoremmentioning

confidence: 99%

Independent sets in hypergraphs

Balogh

Morris

Samotij

2014

J. Amer. Math. Soc.

Self Cite

310

676

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Abstract. Many important theorems and conjectures in combinatorics, such as the theorem of Szemerédi on arithmetic progressions and the Erdős-Stone Theorem in extremal graph theory, can be phrased as statements about families of independent sets in certain uniform hypergraphs. In recent years, an important trend in the area has been to extend such classical results to the so-called 'sparse random setting'. This line of research has recently culminated in the breakthroughs of Conlon and Gowers and of Schacht, who developed general tools for solving problems of this type. Although these two papers solved very similar sets of longstanding open problems, the methods used are very different from one another and have different strengths and weaknesses.In this paper, we provide a third, completely different approach to proving extremal and structural results in sparse random sets that also yields their natural 'counting' counterparts. We give a structural characterization of the independent sets in a large class of uniform hypergraphs by showing that every independent set is almost contained in one of a small number of relatively sparse sets. We then derive many interesting results as fairly straightforward consequences of this abstract theorem. In particular, we prove the wellknown conjecture of Kohayakawa, Luczak, and Rödl, a probabilistic embedding lemma for sparse graphs, for all 2-balanced graphs. We also give alternative proofs of many of the results of Conlon and Gowers and Schacht, such as sparse random versions of Szemerédi's theorem, the Erdős-Stone Theorem and the Erdős-Simonovits Stability Theorem, and obtain their natural 'counting' versions, which in some cases are considerably stronger. We also obtain new results, such as a sparse version of the Erdős-Frankl-Rödl Theorem on the number of H-free graphs and, as a consequence of the K LR conjecture, we extend a result of Rödl and Ruciński on Ramsey properties in sparse random graphs to the general, non-symmetric setting. Similar results have been discovered independently by Saxton and Thomason.

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“…graphs H. When the order of magnitude of ex(n, H) is known then the situation is better: see [5] and [6] for when H is a complete bipartite graph, and see [17] for when H is an even cycle.…”

Section: Denote the Set Of Win Vertices By Wmentioning

confidence: 99%

Recent Trends in Combinatorics

Beveridge¹,

Griggs²,

Hogben³

et al. 2016

The IMA Volumes in Mathematics and Its Applications

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Recently, Balogh-Morris-Samotij and Saxton-Thomason proved that hypergraphs satisfying some natural conditions have only few independent sets. Their main results already have several applications. However, the methods of proving these theorems are even more far reaching. The general idea is to describe some family of events, whose cardinality a priori could be large, only with a few certificates. Here, we show some applications of the methods, including counting C 4 -free graphs, considering the size of a maximum C 4 -free subgraph of a random graph and counting metric spaces with a given number of points. Additionally, we discuss some connections with the Szemerédi Regularity Lemma.

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The number of K m,m -free graphs

Cited by 45 publications

References 18 publications

Galaxy Zoo: the dependence of morphology and colour on environment

Galaxy Zoo: the dependence of morphology and colour on environment

Independent sets in hypergraphs

Recent Trends in Combinatorics

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