arXiv subjects
Publications and source records attributed to Janos Korner.
The zero-error capacity of channels with a countably infinite input alphabet formally generalises Shannon's classical problem about the capacity of discrete memoryless channels. We solve the problem for three particular channels. Our results are purely combinatorial and in line with previous work of the third author about permutation capacity.
We determine the asymptotics of the largest cardinality of a set of Hamilton paths in the complete graph with vertex set [n] under the condition that for any two of the paths in the family there is a subpath of length k entirely contained in only one of them and edge{disjoint from the other one.
We improve by an exponential factor the lower bound of Korner and Muzi for the cardinality of the largest family of Hamilton paths in a complete graph of n vertices in which the union of any two paths has degree 4. The improvement is through an explicit construction while the previous bound was obtained by a greedy algorithm. We solve a similar problem for permutations up to an exponential factor.
We introduce a new class of problems lying halfway between questions about graph capacity and intersection. We say that two binary sequences x and y of the same length have a skewincidence if there is a coordinate i for which x_i=y_{i+1}=1 or vice versa. We give rather sharp bounds on the maximum number of binary sequences of length n any pair of which has a skewincidence.
We extend several results of the third author and C. Malvenuto on graph-different permutations to the case of directed graphs and introduce new open problems. Permutation capacity is a natural extension of Sperner capacity from finite directed graphs to infinite digraphs. Our subject is combinatorial in nature, but can be equally regarded as zero-error information theory.