arXiv · math/0508043
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Anders Bjorner, Richard Stanley. 2006-11-01. An analogue of Young's lattice for compositions. https://arxiv.org/abs/math/0508043
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In 2025 Cornet, Dravec, and Torres determined the domination number $γ(J(n,3))$ of the Johnson graph for every even $n \ge 6$, expressing it as a closed form $ϕ_n$ in terms of Fort-Hedlund covering numbers, and conjectured the same value for odd $n$. We prove this conjecture: $γ(J(n,3)) = ϕ_n$ for every odd $n \ge 7$. A dominating set of $J(n,3)$ is the same as a lotto design $\mathrm{LD}(n,3,3,2)$, and the lotto numbers $L(n,3,3,2)$, and hence $γ(J(n,3))$, were already determined for every $n$ by Bate (1978) and by Brouwer (1981); a short degree argument of Füredi, Székely, and Zubor (1996) gives the same lower bound for all $n \not\equiv 9, 10 \pmod{12}$. The contribution of this paper is therefore a new proof of the odd case, carried out within the bijective framework of Cornet, Dravec, and Torres.
Hales-Jewett theorem is well known. We prove a primitive recursive bound, in fact an $\clE^5$ one, for the density version. In preparation we deal with what was the main induction step in the old proof of Hales-Jewett as a result in its own right with better bounds. Also in improving our bounds.
Let $G$ be a graph with adjacency matrix $A$ and degree matrix $D$. Applying $A$ and $D$ repeatedly to the all-ones vector produces a space $M_G$ of functions on the vertices, and these functions are constant on the orbits of the automorphism group. We call $G$ AD-observable if $M_G$ contains the indicator vector of every orbit, and AD-separable if the functions in $M_G$ separate the orbits. Forests are AD-observable, by McKay's leaf-pruning argument. Our main result is that for every cograph, $M_G$ is spanned by the indicator vectors of the cells of the stable partition of color refinement. Hence a cograph is AD-observable, or equivalently AD-separable, exactly when color refinement finds its orbits, and we describe these cographs by their cotrees. We also show that two cographs with the same number of vertices and the same homomorphism counts from all caterpillars have the same counts from all trees. The main result does not extend to distance-hereditary graphs: we give two such graphs on nine vertices that are AD-separable but not AD-observable.