关于一些有趣的三元公式
On some interesting ternary formulas
摘要：
我们得到以下关于避免三元公式的结果。直到字母的重命名，唯一避免公式$ABCAB.ABCBA.ACB.BAC$的无限三进制字（分别是$ABCA.BCAB.BCB.CBA$）具有与$\texttt{0}\mapsto\texttt{012}$、$\texttt{1}\mapsto\texttt{02}$、$\texttt{2}\mapsto\texttt{1}$的不动点相同的递归因子集。公式$ABAC.BACA.ABCA$被多项式多个二进制字所避免，并且存在任意多个无限二进制字，它们具有避免它的不同递归因子集合。如果一个三元公式的每个变量在同一片段中至少出现两次，则该公式是$3$ -可避免的。对于具有最大度$3$的无$C_4$ -子图类，模式$ABACADABCA$是不可避免的.这就否定了Grytczuk的一个猜想。公式$ABCA.ACBA$，或等价的回文模式$ABCADACBA$，具有可避免性指数$4$。
英文摘要：
We obtain the following results about the avoidance of ternary formulas. Up to renaming of the letters, the only infinite ternary words avoiding the formula $ABCAB.ABCBA.ACB.BAC$ (resp. $ABCA.BCAB.BCB.CBA$) have the same set of recurrent factors as the fixed point of $\texttt{0}\mapsto\texttt{012}$, $\texttt{1}\mapsto\texttt{02}$, $\texttt{2}\mapsto\texttt{1}$. The formula $ABAC.BACA.ABCA$ is avoided by polynomially many binary words and there exist arbitrarily many infinite binary words with different sets of recurrent factors that avoid it. If every variable of a ternary formula appears at least twice in the same fragment, then the formula is $3$-avoidable. The pattern $ABACADABCA$ is unavoidable for the class of $C_4$-minor-free graphs with maximum degree~$3$. This disproves a conjecture of Grytczuk. The formula $ABCA.ACBA$, or equivalently the palindromic pattern $ABCADACBA$, has avoidability index $4$.
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