# Independence numbers of locally sparse graphs and a Ramsey type problem

@article{Alon1996IndependenceNO, title={Independence numbers of locally sparse graphs and a Ramsey type problem}, author={N. Alon}, journal={Random Struct. Algorithms}, year={1996}, volume={9}, pages={271-278} }

Let G = (V,E) be a graph on n vertices with average degree t ≥ 1 in which for every vertex v ∈ V the induced subgraph on the set of all neighbors of v is r-colorable. We show that the independence number of G is at least c log (r+1) n t log t, for some absolute positive constant c. This strengthens a well known result of Ajtai, Komlós and Szemerédi. Combining their result with some probabilistic arguments, we prove the following Ramsey type theorem, conjectured by Erdös in 1979. There exists an… Expand

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#### References

SHOWING 1-8 OF 8 REFERENCES

The average size of an independent set in graphs with a given chromatic number

- Computer Science
- J. Comb. Theory, Ser. B
- 1988

Even if all these small subsets of vertices are independent, still the average size of an independent set in G satisfies (1), and Linial and Saks asked if (1) can be improved. Expand

On the Independence Number of Sparse Graphs

- Mathematics, Computer Science
- Random Struct. Algorithms
- 1995

This work shows for large d that α ≥ c(r)n, the independence number of G, a regular graph of degree d on n points which contains no Kr (r ≥ 4). Expand

A note on the independence number of triangle-free graphs

- Computer Science, Mathematics
- Discret. Math.
- 1983

This note gives a simple proof that @a >= n (d ln d - d + 1)/(d - 1)^2 and considers what happens when G contains a limited number of triangles. Expand

On Turán’s theorem for sparse graphs

- Computer Science, Mathematics
- Comb.
- 1981

This work improves the bound for graphs containing no large cliques by improving Turán’s theorem, which states that the maximum size of an independent set in a graph is the sum of its vertices and valency. Expand

A Note on Ramsey Numbers

- Computer Science, Mathematics
- J. Comb. Theory, Ser. A
- 1980

Upper bounds are found for the Ramsey function and it is proved that R(3, x) cx 2 ln x and, for each k ⩾ 3, R(k,x) c k x k − 1 ( lN x) k − 2 asymptotically in x . Expand

The Probabilistic Method

- Computer Science
- SODA
- 1992

A particular set of problems - all dealing with “good” colorings of an underlying set of points relative to a given family of sets - is explored. Expand

Ramsey Theory (Second Edition)

- Wiley, New York
- 1990