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Register a new userA Hales--Jewett type property of finite solvable groups
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A conjecture of Leader, Russell and Walters in Euclidean Ramsey theory says that a finite set is Ramsey if and only if it is congruent to a subset of a set whose symmetry group acts transitively. As they have shown the ``if direction of their conjecture follows if all finite groups have a Hales--Jewett type property. In this paper, we show that this property is satisfied in the case of finite solvable groups. Our result can be used to recover the work of Kv{r}iv{z} in Euclidean Ramsey theory.
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The Hales-Jewett theorem for alphabet of size 3 states that whenever the Hales-Jewett cube [3]^n is r-coloured there is a monochromatic line (for n large). Conlon and Kamcev conjectured that, for any n, there is a 2-colouring of [3]^n for which there is no monochromatic line whose active coordinate set is an interval. In this note we disprove this conjecture.
For integers $n$ and $k$, the density Hales-Jewett number $c_{n,k}$ is defined as the maximal size of a subset of $[k]^n$ that contains no combinatorial line. We show that for $k ge 3$ the density Hales-Jewett number $c_{n,k}$ is equal to the maximal size of a cylinder intersection in the problem $Part_{n,k}$ of testing whether $k$ subsets of $[n]$ form a partition. It follows that the communication complexity, in the Number On the Forehead (NOF) model, of $Part_{n,k}$, is equal to the minimal size of a partition of $[k]^n$ into subsets that do not contain a combinatorial line. Thus, the bound in cite{chattopadhyay2007languages} on $Part_{n,k}$ using the Hales-Jewett theorem is in fact tight, and the density Hales-Jewett number can be thought of as a quantity in communication complexity. This gives a new angle to this well studied quantity. As a simple application we prove a lower bound on $c_{n,k}$, similar to the lower bound in cite{polymath2010moser} which is roughly $c_{n,k}/k^n ge exp(-O(log n)^{1/lceil log_2 krceil})$. This lower bound follows from a protocol for $Part_{n,k}$. It is interesting to better understand the communication complexity of $Part_{n,k}$ as this will also lead to the better understanding of the Hales-Jewett number. The main purpose of this note is to motivate this study.
We explore graph theoretical properties of minimal prime graphs of finite solvable groups. In finite group theory studying the prime graph of a group has been an important topic for the past almost half century. Recently prime graphs of solvable groups have been characterized in graph theoretical terms only. This now allows the study of these graphs with methods from graph theory only. Minimal prime graphs turn out to be of particular interest, and in this paper we pursue this further by exploring, among other things, diameters, Hamiltonian cycles and the property of being self-complementary for minimal prime graphs. We also study a new, but closely related notion of minimality for prime graphs and look into counting minimal prime graphs.
A Cayley graph is said to be an NNN-graph if it is both normal and non-normal for isomorphic regular groups, and a group has the NNN-property if there exists an NNN-graph for it. In this paper we investigate the NNN-property of cyclic groups, and show that cyclic groups do not have the NNN-property.
We lay down the fundations of the theory of groups of finite Morley rank in which local subgroups are solvable and we proceed to the local analysis of these groups. We prove the main Uniqueness Theorem, analogous to the Bender method in finite group theory, and derive its corollaries. We also consider homogeneous cases as well as torsion.
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