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Register a new userA Dichotomy Theorem for First-Fit Chain Partitions
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First-Fit is a greedy algorithm for partitioning the elements of a poset into chains. Let $textrm{FF}(w,Q)$ be the maximum number of chains that First-Fit uses on a $Q$-free poset of width $w$. A result due to Bosek, Krawczyk, and Matecki states that $textrm{FF}(w,Q)$ is finite when $Q$ has width at most $2$. We describe a family of posets $mathcal{Q}$ and show that the following dichotomy holds: if $Qinmathcal{Q}$, then $textrm{FF}(w,Q) le 2^{c(log w)^2}$ for some constant $c$ depending only on $Q$, and if $Q otinmathcal{Q}$, then $textrm{FF}(w,Q) ge 2^w - 1$.
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For an integer $mge 2$, a partition $lambda=(lambda_1,lambda_2,ldots)$ is called $m$-falling, a notion introduced by Keith, if the least nonnegative residues mod $m$ of $lambda_i$s form a nonincreasing sequence. We extend a bijection originally due to the third author to deduce a lecture hall theorem for such $m$-falling partitions. A special case of this result gives rise to a finite version of Pak-Postnikovs $(m,c)$-generalization of Eulers theorem. Our work is partially motivated by a recent extension of Eulers theorem for all moduli, due to Keith and Xiong. We note that their result actually can be refined with one more parameter.
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A seminal theorem of Tverberg states that any set of $T(r,d)=(r-1)(d+1)+1$ points in $mathbb{R}^d$ can be partitioned into $r$ subsets whose convex hulls have non-empty $r$-fold intersection. Almost any collection of fewer points in $mathbb{R}^d$ cannot be so divided, and in these cases we ask if the set can nonetheless be $P(r,d)$--partitioned, i.e., split into $r$ subsets so that there exist $r$ points, one from each resulting convex hull, which form the vertex set of a prescribed convex $d$--polytope $P(r,d)$. Our main theorem shows that this is the case for any generic $T(r,2)-2$ points in the plane and any $rgeq 3$ when $P(r,2)=P_r$ is a regular $r$--gon, and moreover that $T(r,2)-2$ is tight. For higher dimensional polytopes and $r=r_1cdots r_k$, $r_i geq 3$, this generalizes to $T(r,2k)-2k$ generic points in $mathbb{R}^{2k}$ and orthogonal products $P(r,2k)=P_{r_1}times cdots times P_{r_k}$ of regular polygons, and likewise to $T(2r,2k+1)-(2k+1)$ points in $mathbb{R}^{2k+1}$ and the product polytopes $P(2r,2k+1)=P_{r_1}times cdots times P_{r_k} times P_2$. As with Tverbergs original theorem, our results admit topological generalizations when $r$ is a prime power, and, using the constraint method of Blagojevic, Frick, and Ziegler, allow for dimensionally restrict
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