Low sets without subsets of higher many-one degree

Given a reducibility $\leq_\mathrm{r}$, we say that an infinite set $A$ is $r$-introimmune if $A$ is not $r$-reducible to any of its subsets $B$ with $|A\backslash B|=\infty$. We consider the many-one reducibility $\leq_\mathrm{m}$ and we prove the existence of a low$_1$ $m$-introimmune set in $\Pi^0_1$ and the existence of a low$_1$ bi-$m$-introimmune set

Colouring set families without monochromatic k-chains

A coloured version of classic extremal problems dates back to Erd\H{o}s and Rothschild, who in 1974 asked which $n$-vertex graph has the maximum number of 2-edge-colourings without monochromatic triangles. They conjectured that the answer is simply given by the largest triangle-free graph. Since then, this new class of coloured extremal problems has been extensively studied by various researchers. In this paper we pursue the Erd\H{o}s--Rothschild versions of Sperner's Theorem, the classic result in extremal set theory on the size of the largest antichain in the Boolean lattice, and Erd\H{o}s' extension to $k$-chain-free families. Given a family $\mathcal{F}$ of subsets of $[n]$, we define an $(r,k)$-colouring of $\mathcal{F}$ to be an $r$-colouring of the sets without any monochromatic $k$-chains $F_1 \subset F_2 \subset \dots \subset F_k$. We prove that for $n$ sufficiently large in terms of $k$, the largest $k$-chain-free families also maximise the number of $(2,k)$-colourings. We also show that the middle level, $\binom{[n]}{\lfloor n/2 \rfloor}$, maximises the number of $(3,2)$-colourings, and give asymptotic results on the maximum possible number of $(r,k)$-colourings whenever $r(k-1)$ is divisible by three.Comment: 30 pages, final versio

Design agents and the need for high-dimensional perception

Designed artefacts may be quantified by any number of measures. This paper aims to show that in doing so, the particular measures used may matter very little, but as many as possible should be taken. A set of building plans is used to demonstrate that arbitrary measures of their shape serve to classify them into neighbourhood types, and the accuracy of classification increases as more are used, even if the dimensionality of the space in which classification occurs is held constant. It is further shown that two autonomous agents may independently choose sets of attributes by which to represent the buildings, but arrive at similar judgements as more are used. This has several implications for studying or simulating design. It suggests that quantitative studies of collections of artefacts may be made without requiring extensive knowledge of the best possible measures—often impossible in real, ill-defined, design situations. It suggests a means by which the generation of novelty can be explained in a group of agents with different ways of seeing a given event. It also suggests that communication can occur without the need for predetermined codes or protocols, introducing the possibility of alternative human-computer interfaces that may be useful in design

Overlap properties of geometric expanders

The {\em overlap number} of a finite $(d+1)$-uniform hypergraph $H$ is defined as the largest constant $c(H)\in (0,1]$ such that no matter how we map the vertices of $H$ into $\R^d$, there is a point covered by at least a $c(H)$-fraction of the simplices induced by the images of its hyperedges. In~\cite{Gro2}, motivated by the search for an analogue of the notion of graph expansion for higher dimensional simplicial complexes, it was asked whether or not there exists a sequence $\{H_n\}_{n=1}^\infty$ of arbitrarily large $(d+1)$-uniform hypergraphs with bounded degree, for which $\inf_{n\ge 1} c(H_n)>0$. Using both random methods and explicit constructions, we answer this question positively by constructing infinite families of $(d+1)$-uniform hypergraphs with bounded degree such that their overlap numbers are bounded from below by a positive constant $c=c(d)$. We also show that, for every $d$, the best value of the constant $c=c(d)$ that can be achieved by such a construction is asymptotically equal to the limit of the overlap numbers of the complete $(d+1)$-uniform hypergraphs with $n$ vertices, as $n\rightarrow\infty$. For the proof of the latter statement, we establish the following geometric partitioning result of independent interest. For any $d$ and any $\epsilon>0$, there exists $K=K(\epsilon,d)\ge d+1$ satisfying the following condition. For any $k\ge K$, for any point $q \in \mathbb{R}^d$ and for any finite Borel measure $\mu$ on $\mathbb{R}^d$ with respect to which every hyperplane has measure $0$, there is a partition $\mathbb{R}^d=A_1 \cup \ldots \cup A_{k}$ into $k$ measurable parts of equal measure such that all but at most an $\epsilon$-fraction of the $(d+1)$-tuples $A_{i_1},\ldots,A_{i_{d+1}}$ have the property that either all simplices with one vertex in each $A_{i_j}$ contain $q$ or none of these simplices contain $q$