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Keyword: extremal set theory ×Clear all

3 claims from 1 paper

1. Mathematics › Limits and Structures in Graph Theory

   On generalized Ramsey numbers of Erdős and Rogers

   Dudek, Retter and Rödl · Journal of Combinatorial Theory Series B · 2014

   The paper bounds the Erdős–Rogers function, giving an upper bound for f_{s,s+1}(n) up to polylogarithmic factors and a square-root bound for f_{s,t}(n) when t−2 ≥ s ≥ 4.

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   1. Unchecked“In this paper, we show that for every $s\ge 3$ there exist constants $c_1=c_1(s)$ and $c_2=c_2(s)$ such that $f_{s,s+1}(n) \le c_1 (\log n)^{c_2} \sqrt{n}$.”
   2. UncheckedFor t−2 ≥ s ≥ 4, the Erdős–Rogers function f_{s,t}(n) is bounded above by a constant times the square root of n.“We also show for all $t-2 \geq s \geq 4$, there exists a constant $c_3$ such that $f_{s,t}(n) \le c_3 \sqrt{n}$.”
   3. UncheckedFor every s at least 4, the ratio f_{s+1,s+2}(n) to f_{s,s+2}(n) grows without bound as n grows, partially answering a question of Erdős.“In doing so, we partially answer a question of Erdős by showing that $\lim_{n\to \infty} \frac{f_{s+1,s+2}(n)}{f_{s,s+2}(n)}=\infty$ for any $s\ge 4$.”