14 research outputs found
New Examples of Kochen-Specker Type Configurations on Three Qubits
A new example of a saturated Kochen-Specker (KS) type configuration of 64
rays in 8-dimensional space (the Hilbert space of a triple of qubits) is
constructed. It is proven that this configuration has a tropical dimension 6
and that it contains a critical subconfiguration of 36 rays. A natural
multicolored generalisation of the Kochen-Specker theory is given based on a
concept of an entropy of a saturated configuration of rays.Comment: 24 page
Internal Partitions of Regular Graphs
An internal partition of an -vertex graph is a partition of
such that every vertex has at least as many neighbors in its own part as in the
other part. It has been conjectured that every -regular graph with
vertices has an internal partition. Here we prove this for . The case
is of particular interest and leads to interesting new open problems on
cubic graphs. We also provide new lower bounds on and find new families
of graphs with no internal partitions. Weighted versions of these problems are
considered as well
Exact big Ramsey degrees via coding trees
We characterize the big Ramsey degrees of free amalgamation classes in finite
binary languages defined by finitely many forbidden irreducible substructures,
thus refining the recent upper bounds given by Zucker. Using this
characterization, we show that the Fra\"iss\'e limit of each such class admits
a big Ramsey structure. Consequently, the automorphism group of each such
Fra\"iss\'e limit has a metrizable universal completion flow.Comment: Submitted versio