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frafour
frafour
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At least one of the questions admit counterexamples. Namely, let $\Sigma = \{a, b, c, d \}$, $I = \{ ab^nc : n \geq 1\}$ and $S = \{ p d s : p, s \in I; p \neq s \}$$S = \{ ab^nc d ab^mc : n > m \}$. Then all words in $S$ are unborderedLyndon, $OG(S)$ has clique number equal to $3$is triangle-free and has infinite chromatic number. I can give more details if anyone is interested.

At least one of the questions admit counterexamples. Namely, let $\Sigma = \{a, b, c, d \}$, $I = \{ ab^nc : n \geq 1\}$ and $S = \{ p d s : p, s \in I; p \neq s \}$. Then all words in $S$ are unbordered, $OG(S)$ has clique number equal to $3$ and infinite chromatic number. I can give more details if anyone is interested.

At least one of the questions admit counterexamples. Namely, let $\Sigma = \{a, b, c, d \}$, and $S = \{ ab^nc d ab^mc : n > m \}$. Then all words in $S$ are Lyndon, $OG(S)$ is triangle-free and has infinite chromatic number. I can give more details if anyone is interested.

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frafour
frafour
  • 435
  • 2
  • 8

At least one of the questions admit counterexamples. Namely, let $\Sigma = \{a, b, c, d \}$, $I = \{ ab^nc : n \geq 1\}$ and $S = \{ p d s : p, s \in I; p \neq s \}$. Then all words in $S$ are unbordered, $OG(S)$ has clique number equal to $3$ and infinite chromatic number. I can give more details if anyone is interested.