AviKndr/universal-words
Answer to Problem 10.32 of the Kourovka Notebook, posed by Yu. I. Merzlyakov in 1986 and still open in the Notebook's 21st edition (2026), in both directions. Positive direction: there are constants C and N0 such that for all nonzero integers r, s with r or s odd, the word x^r y^s is universal on the symmetric group S_n (every permutation is a value) whenever n >= N0 and n >= C * log m(r,s), where m(r,s) is the product of the distinct primes dividing rs. This is the S_n analogue of the logarithmic Brenner-Evans-Silberger bound for A_n; the best previous S_n bound (Droste, 1986) was linear in…
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