Rainbow Turán number of even cycles, repeated patterns and blow-ups of cycles

Janzer, Oliver

doi:10.48550/arxiv.2006.01062

Cited by 11 publications

(18 citation statements)

References 16 publications

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“…We remark that our method is very different from the approach of O. Janzer [18], it is closer in spirit to that of [8].…”

Section: Rainbow Cycles and Subdivisionsmentioning

confidence: 79%

“…Coloring an edge with color i if its end-vertices differ in coordinate i gives a proper coloring with no rainbow cycles. On the other hand, Das, Lee, and Sudakov [8] proved that average degree e (log n) 1/2+o(1) guarantees a rainbow cycle, which was further improved to O((log n) 4 ) by O. Janzer [18]. By studying properly edge colored α-maximal graphs with α ≈ 1/ log n, we are able to get within a log factor of the lower bound.…”

Section: Rainbow Cycles and Subdivisionsmentioning

confidence: 89%

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Robust (rainbow) subdivisions and simplicial cycles

Tomon¹

2022

Preprint

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We present several results in extremal graph and hypergraph theory of topological nature. First, we show that if α > 0 and ℓ = Ω( 1 α log 1 α ) is an odd integer, then every graph G with n vertices and at least n 1+α edges contains an ℓ-subdivision of the complete graph K t , where t = n Θ(α) . Also, this remains true if in addition the edges of G are properly colored, and one wants to find a rainbow copy of such a subdivision. In the sparser regime, we show that properly edge colored graphs on n vertices with average degree (log n) 2+o(1) contain rainbow cycles, while average degree (log n) 6+o(1) guarantees rainbow subdivisions of K t for any fixed t, thus improving recent results of Janzer and Jiang et al., respectively. Furthermore, we consider certain topological notions of cycles in pure simplicial complexes (uniform hypergraphs). We show that if G is a 2-dimensional pure simplicial complex (3-graph) with n 1-dimensional and at least n 1+α 2-dimensional faces, then G contains a triangulation of the cylinder and the Möbius strip with O( 1 α log 1 α ) vertices. We present generalizations of this for higher dimensional pure simplicial complexes as well.In order to prove these results, we consider certain (properly edge colored) graphs and hypergraphs G with strong expansion. We argue that if one randomly samples the vertices (and colors) of G with not too small probability, then many pairs of vertices are connected by a short path whose vertices (and colors) are from the sampled set, with high probability.

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“…We remark that our method is very different from the approach of O. Janzer [18], it is closer in spirit to that of [8].…”

Section: Rainbow Cycles and Subdivisionsmentioning

confidence: 79%

Section: Rainbow Cycles and Subdivisionsmentioning

confidence: 89%

Robust (rainbow) subdivisions and simplicial cycles

Tomon¹

2022

Preprint

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“…) where ε k → ∞ as k → ∞. Very recently, Janzer [3] proved the conjecture by showing that ex * (n, C 2k ) = O(n 1+1/k ).…”

Section: Introductionmentioning

confidence: 92%

“…Theorem 1.2 (Janzer [3]). There is a constant C such that if n is sufficiently large and G is a properly edge-coloured n-vertex graph with at least Cn(log n) 4 edges, then G contains a rainbow cycle of even length.…”

Section: Introductionmentioning

confidence: 99%

Rainbow Turán number of clique subdivisions

Jiang,

Methuku,

Yepremyan

2021

Preprint

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We show that for any integer t ≥ 2, every properly edge-coloured graph on n vertices with more than n 1+o(1) edges contains a rainbow subdivision of K t . Note that this bound on the number of edges is sharp up to the o(1) error term. This is a rainbow analogue of some classical results on clique subdivisions and extends some results on rainbow Turán numbers. Our method relies on the framework introduced by Sudakov and Tomon [10] which we adapt to find robust expanders in the coloured setting.

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“…Applying (4) gives a result which matches Theorem 2.3 due to our current knowledge of the extremal number of K + t . Among other impressive results, Janzer [12] showed that for fixed integers k, r ≥ 2,…”

Section: Relation To the Conlon-tyomkyn Problemmentioning

confidence: 99%

Lower bounds on the Erdős-Gyárfás problem via color energy graphs

Balogh¹,

English²,

Heath³

et al. 2021

Preprint

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Given positive integers p and q, a (p, q)-coloring of the complete graph K n is an edge-coloring in which every p-clique receives at least q colors. Erdős and Shelah posed the question of determining f (n, p, q), the minimum number of colors needed for a (p, q)-coloring of K n . In this paper, we expand on the color energy technique introduced by Pohoata and Sheffer to prove new lower bounds on this function, making explicit the connection between bounds on extremal numbers and f (n, p, q). Using results on the extremal numbers of subdivided complete graphs, theta graphs, and subdivided complete bipartite graphs, we generalize results of Fish, Pohoata, and Sheffer, giving the first nontrivial lower bounds on f (n, p, q) for some pairs (p, q) and improving previous lower bounds for other pairs.

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Rainbow Turán number of even cycles, repeated patterns and blow-ups of cycles

Cited by 11 publications

References 16 publications

Robust (rainbow) subdivisions and simplicial cycles

Robust (rainbow) subdivisions and simplicial cycles

Rainbow Turán number of clique subdivisions

Lower bounds on the Erdős-Gyárfás problem via color energy graphs

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