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For a finite real alphabet $T$, let $g(T)$ be the supremum of the lengths of words over $T$ containing no two adjacent nonempty blocks of equal length and equal sum. We classify all four-element alphabets with $g(T)<139$. Up to affine equivalence, these are the known endpoint-sum family and seven isolated classes. Exha...
For every $n\ge9$, we construct a strongly connected synchronizing automaton with two input letters and $n$ states whose shortest word merging some three distinct states has length $\lfloor4n/3\rfloor$. One letter is a permutation; the other has an image of size $n-1$. We give complete transition maps and an explicit w...
We study simultaneous color densities in packing colorings of integer distance graphs. For $D(1,6)$, we determine several exact density regions and prove that colors $1$ through $7$ have maximum combined density $211/252$. When this maximum is approached, the seven individual color frequencies are forced to converge to...
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