Improved bounds for the Erdős-Rogers function

Gowers, Timothy; Janzer, Oliver

doi:10.19086/aic.12048

Cited by 2 publications

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“…Throughout the years, the upper bounds and lower bounds on f s,s+t (n) for different pairs (s, t) have been extensively studied (see [1,8,9,10,11,4,15,3,7]). The case t = s + 1 has received the most attention.…”

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confidence: 99%

On the stability of graph independence number

Dong,

2021

Preprint

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Let G be a graph on n vertices of independence number α(G) such that every induced subgraph of G on n − k vertices has an independent set of size at least α(G) − ℓ. What is the largest possible α(G) in terms of n for fixed k and ℓ? We show that α(G)We also use this result to determine new values of the Erdős-Rogers function. IntroductionBackground. All graphs considered here are finite, undirected, and simple. For graph G and property P, the resilience of P measures how much one should change G in order to destroy P.Assume G has P, then the global resilience of P refers to the minimum number r such that by removing r edges from G one can obtain a graph not having P, and the local resilience refers to the minimum number r such that by removing at each vertex at most r edges one can obtain a graph not having P. For example, Turán's theorem (see [13]) characterizes the global resilience of having a k-clique in complete graphs, and Dirac's theorem (see [2]) characterizes the local resilience of having a Hamiltonian path in complete graphs. Moreover, the local resilience of various properties is extensively studied in [12].Note that both global resilience and local resilience focus on removing edges. What about removing vertices? As far as we are aware of, this vertex-removal version of resilience is never discussed. To distinguish from removing edges, we shall always use the word stability when discussing removing vertices throughout this paper.

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On the stability of graph independence number

Dong,

2021

Preprint

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“…Choosing r = 2 precisely recovers the usual Ramsey problem and was in fact the original motivation behind the general question. This question became known as the Erdős-Rogers problem and has been extensively studied, for some examples see [9,11,13,20,25,[37][38][39] and a recent survey [10] due to Dudek and Rödl.…”

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Large Independent Sets from Local Considerations

Bucić

Sudakov

2023

Combinatorica

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The following natural problem was raised independently by Erdős–Hajnal and Linial–Rabinovich in the early ’90 s. How large must the independence number $$\alpha (G)$$ α ( G ) of a graph G be whose every m vertices contain an independent set of size r? In this paper, we discuss new methods to attack this problem. The first new approach, based on bounding Ramsey numbers of certain graphs, allows us to improve the previously best lower bounds due to Linial–Rabinovich, Erdős–Hajnal and Alon–Sudakov. As an example, we prove that any n-vertex graph G having an independent set of size 3 among every 7 vertices has $$\alpha (G) \ge \Omega (n^{5/12})$$ α ( G ) ≥ Ω ( n 5 / 12 ) . This confirms a conjecture of Erdős and Hajnal that $$\alpha (G)$$ α ( G ) should be at least $$n^{1/3+\varepsilon }$$ n 1 / 3 + ε and brings the exponent halfway to the best possible value of 1/2. Our second approach deals with upper bounds. It relies on a reduction of the original question to the following natural extremal problem. What is the minimum possible value of the 2-density (The 2-density of a graph H is defined as $$m_2(H):=\max _{H' \subseteq H, |H'| \ge 3} \frac{e(H')-1}{|H'|-2}$$ m 2 ( H ) : = max H ′ ⊆ H , | H ′ | ≥ 3 e ( H ′ ) - 1 | H ′ | - 2 ) of a graph on m vertices having no independent set of size r? This allows us to improve previous upper bounds due to Linial–Rabinovich, Krivelevich and Kostochka-Jancey. As part of our arguments, we link the problem of Erdős–Hajnal and Linial–Rabinovich and our new extremal 2-density problem to a number of other well-studied questions. This leads to many interesting directions for future research.

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Improved bounds for the Erdős-Rogers function

Cited by 2 publications

References 13 publications

On the stability of graph independence number

On the stability of graph independence number

Large Independent Sets from Local Considerations

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