In this paper, we study Condorcet domains, sets of linear orders from which majority ranking produces a linear order. We introduce a new class of Condorcet domains, called coherent domains, which is natural from both a voting theoretic and combinatorial perspective. After studying the properties of these domains we introduce set-alternating schemes. This is a method for constructing well-behaved coherent domains. Using this we show that, for sufficiently large numbers of alternatives n , there are coherent domains of size more than 2 . 197 3 n . This improves the best existing asymptotic lower bounds for the size of the largest general Condorcet domains.

Let $H_n$ be a $k$-graph on $n$ vertices. For $0 \le \ell \ell >1$, the set of $\pi_{\ell}^k(\mathcal{F})$ is dense in the interval $[0,1)$. Hence, there is no “jump” for $\ell$-degree Turán density when $k>\ell >1$. We also give a lower bound on $\pi_{\ell}^k(\mathcal{F})$ in terms of an ordinary Turán density.

The notion of a 1-vertex transfer matrix for multidimensional codes is introduced. It is shown that the capacity of such codes, or the topological entropy, can be expressed as the limit of the logarithm of spectral radii of 1-vertex transfer matrices. Storage and computations using the 1-vertex transfer matrix are much smaller than storage and computations needed for the standard transfer matrix. The method is applied to estimate the first 15 digits of the entropy of the 2-D(0, 1)run length limited channel. A large-scale computation of eigenvalues for the(0, 1)run length limited channel in 2-D and 3-D have been carried out. This was done in order to be able to compare the computational cost of the new method with the standard transfer matrix and have rigorous bounds to compare the estimates with. This in turn leads to improvements on the best previous lower and upper bounds for these channels.