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Register a new userA geometric Hall-type theorem
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We introduce a geometric generalization of Halls marriage theorem. For any family $F = {X_1, dots, X_m}$ of finite sets in $mathbb{R}^d$, we give conditions under which it is possible to choose a point $x_iin X_i$ for every $1leq i leq m$ in such a way that the points ${x_1,...,x_m}subset mathbb{R}^d$ are in general position. We give two proofs, one elementary proof requiring slightly stronger conditions, and one proof using topological techniques in the spirit of Aharoni and Haxells celebrated generalization of Halls theorem.
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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.
We define and study slider-pinning rigidity, giving a complete combinatorial characterization. This is done via direction-slider networks, which are a generalization of Whiteleys direction networks.
We give a bound on the spectral radius of a graph implying a quantitative version of the Erdos-Stone theorem.
In 1882 J.J. Sylvester already proved, that the number of different ways to partition a positive integer into consecutive positive integers exactly equals the number of odd divisors of that integer (see [1]). We will now develop an interesting statement about triangular numbers, those positive integers which can be partitioned into consecutive numbers beginning at 1. For every partition of a triangular number n into consecutive numbers we can partition the sequence of numbers beginning at 1, adding up to n again, such that every part of this partition adds up to exactly one number of the chosen partition of n.
In 1972, Chvatal gave a well-known sufficient condition for a graphical sequence to be forcibly hamiltonian, and showed that in some sense his condition is best possible. Nash-Williams gave examples of forcibly hamiltonian n-sequences that do not satisfy Chvatlas condition for every n at least 5. In this note we generalize the Nash-Williams examples, and use this generalization to generate Omega(2^n/n^.5) forcibly hamiltonian n-sequences that do not satisfy Chvatals condition
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