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We improve the best known upper bound on the length of the shortest resetwords of synchronizing automata. The new bound is slightly better than $114 n^3/ 685 + O(n^2)$. The \v{C}ern\'y conjecture states that $(n-1)^2$ is an upperbound. So far, the best general upper bound was $(n^3-n)/6-1$ obtained byJ.-E.~Pin and P.~Frankl in 1982. Despite a number of efforts, it remainedunchanged for about 35 years.  To obtain the new upper bound we utilize avoiding words. A word is avoidingfor a state $q$ if after reading the word the automaton cannot be in $q$. Weobtain upper bounds on the length of the shortest avoiding words, and using theapproach of Trahtman from 2011 combined with the well known Frankl theorem from1982, we improve the general upper bound on the length of the shortest resetwords. For all the bounds, there exist polynomial algorithms finding a word oflength not exceeding the bound.

Improving the upper bound on the length of the shortest reset words