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Ramsey number
Ramsey properties of Cayley graphs ★★★
Author(s): Alon
Conjecture There exists a fixed constant
so that every abelian group
has a subset
with
so that the Cayley graph
has no clique or independent set of size
.
![$ c $](/files/tex/dccee841f3f498c2c58fa6ae1c1403c5a88c5b8d.png)
![$ G $](/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png)
![$ S \subseteq G $](/files/tex/1443dce39cbce9606ca2c26bfb4ffbcf28ddb2c9.png)
![$ -S = S $](/files/tex/c931921f156211c8e73ee3e57353f799f2676a34.png)
![$ {\mathit Cayley}(G,S) $](/files/tex/f0cecf17b88dfdb3db14ed2c21bce70a47d051cd.png)
![$ > c \log |G| $](/files/tex/e224dcfe2b20fdd16ed3f4a07612dc76b28666fe.png)
Keywords: Cayley graph; Ramsey number
Diagonal Ramsey numbers ★★★★
Author(s): Erdos
Let denote the
diagonal Ramsey number.
Conjecture
exists.
![$ \lim_{k \rightarrow \infty} R(k,k) ^{\frac{1}{k}} $](/files/tex/239db4dc95ea810751ae620dbaa3da745636e4cc.png)
Problem Determine the limit in the above conjecture (assuming it exists).
Keywords: Ramsey number
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