We show that if add(null) = c, then the globally Baire and universally measurable chromatic numbers of the graph of any Borel function on a Polish space are equal and at most three. In particular, this holds for the graph of the unilateral shift on [N]N, although its Borel chromatic number is N₀. We also show that if add(null) = c, then the universally measurable chromatic number of every treeing of a measure amenable equivalence relation (...) is at most three. In particular, this holds for "the" minimum analytic graph G₀ with uncountable Borel (and Baire measurable) chromatic number. In contrast, we show that for all κ ∈ {2, 3...., N₀. c}, there is a treeing of E₀ with Borel and Baire measurable chromatic number κ. Finally, we use a Glimm—Effros style dichotomy theorem to show that every basis for a non-empty initial segment of the class of graphs of Borel functions of Borel chromatic number at least three contains a copy of (R<N, ⊇). (shrink)

Journal of Mathematical Logic, Ahead of Print. We construct bounded degree acyclic Borel graphs with large Borel chromatic number using a graph arising from Ramsey theory and limits of expander sequences.

We construct bounded degree acyclic Borel graphs with large Borel chromatic number using a graph arising from Ramsey theory and limits of expander sequences.

For each infinite cardinalκ, the set of algebraic hypergraphs having chromatic number no larger thanκis decidable.

Given a graph G whose set of vertices is a Polish space X, the weak Borel chromatic number of G is the least size of a family of pairwise disjoint G -independent Borel sets that covers all of X. Here a set of vertices of a graph G is independent if no two vertices in the set are connected by an edge.We show that it is consistent with an arbitrarily large size of the continuum that every closed graph (...) on a Polish space either has a perfect clique or has a weak Borel chromatic number of at most ℵ1. We observe that some weak version of Todorcevic's Open Coloring Axiom for closed colorings follows from MA.Slightly weaker results hold for Fσ-graphs. In particular, it is consistent with an arbitrarily large size of the continuum that every locally countable Fσ-graph has a Borel chromatic number of at most ℵ1.We refute various reasonable generalizations of these results to hypergraphs. (shrink)

We show that there is an antichain basis for neither the class of non-potentially closed Borel subsets of the plane under Borel rectangular reducibility nor the class of analytic graphs of uncountable Borel chromatic number under Borel reducibility.

A recursive graph is a graph whose vertex and edge sets are recursive. A highly recursive graph is a recursive graph that also has the following property: one can recursively determine the neighbors of a vertex. Both of these have been studied in the literature. We consider an intermediary notion: Let A be a set. An A-recursive graph is a recursive graph that also has the following property: one can recursively-in-A determine the neighbors of a vertex. We show that, if (...) A is r.e. and not recursive, then there exists A-recursive graphs that are 2-colorable but not recursively k-colorable for any k. This is false for highly-recursive graphs but true for recursive graphs. Hence A-recursive graphs are closer in spirit to recursive graphs than to highly recursive graphs. (shrink)

The number of discriminable colours is often assumed to be of the order of several million but the extent of detectable chromatic diversity present in individual natural scenes is an open question. Here, the aim was to estimate the number of discriminable colours seen in natural scenes. Hyperspectral data were obtained from a set of natural scenes over the range 400 - 720 nm at 10 nm intervals (Nascimento et al, 2002 Journal of the Optical Society of (...) America A 19 1484 - 1490) and the representation of each scene in CIELAB space computed for all pixels of the image. The number of discriminable colours was estimated by counting the number of unit volumes in the three-dimensional volume defined by the scene when represented in CIELAB space. We found that the number of discriminable colours estimated in a scene could vary by an order of magnitude from scene to scene, but rarely exceeded 105, indicating a limited chromatic diversity. (shrink)

Summary The twofold objective of this study is (1) to identify and give a brief review of the historical development of the various designs of early (pre-1850) telescope eyepieces, and (2) to determine by measurements and calculations the axial and lateral chromatic aberrations of a number of extant eyepieces from that period in order to provide basic data on which to judge the relative quality of different eyepiece forms. Eight distinct types of eyepieces containing one to five lens (...) elements are discussed and illustrated. The second objective was addressed by measuring the focal lengths of the individual lens elements and the lens spacings in each eyepiece. The data were processed by a ray-tracing program that yields the chromatic aberration (CA) and focal length of each of the eyepieces. Twenty-one telescopes from that period were studied in this way. Similar calculations were also made using data available in the literature on several additional historic telescopes. The conclusions drawn from the data are: (1) Telescopes with Galilean eyepieces have a smaller CA than those with Keplerian or Huygens eyepieces. (2) The two-lens Keplerian terrestrial eyepieces are demonstrated to have larger axial and lateral CA than any of the other eyepiece types. (3) Calculations from examples of telescopes by Divini and Campani for which data were available indicate that while the axial CAs of the two were the same, Campani's instrument had a much smaller lateral CA. (4) Five element eyepieces were found to be better corrected for CA than the early four-lens systems that replaced them. (5) This study tends to confirm the speculation that the reason makers of early achromatic telescopes overcorrected the objective lenses was to compensate for the CA of the eyepieces. (shrink)

Summary The twofold objective of this study is (1) to identify and give a brief review of the historical development of the various designs of early (pre-1850) telescope eyepieces, and (2) to determine by measurements and calculations the axial and lateral chromatic aberrations of a number of extant eyepieces from that period in order to provide basic data on which to judge the relative quality of different eyepiece forms. Eight distinct types of eyepieces containing one to five lens (...) elements are discussed and illustrated. The second objective was addressed by measuring the focal lengths of the individual lens elements and the lens spacings in each eyepiece. The data were processed by a ray-tracing program that yields the chromatic aberration (CA) and focal length of each of the eyepieces. Twenty-one telescopes from that period were studied in this way. Similar calculations were also made using data available in the literature on several additional historic telescopes. The conclusions drawn from the data are: (1) Telescopes with Galilean eyepieces have a smaller CA than those with Keplerian or Huygens eyepieces. (2) The two-lens Keplerian terrestrial eyepieces are demonstrated to have larger axial and lateral CA than any of the other eyepiece types. (3) Calculations from examples of telescopes by Divini and Campani for which data were available indicate that while the axial CAs of the two were the same, Campani's instrument had a much smaller lateral CA. (4) Five element eyepieces were found to be better corrected for CA than the early four-lens systems that replaced them. (5) This study tends to confirm the speculation that the reason makers of early achromatic telescopes overcorrected the objective lenses was to compensate for the CA of the eyepieces. (shrink)

Visual percepts frequently appear chromatically rich, yet their paucity in reportable information has led to widely accepted minimalist models of vision. The discrepancy may be resolved by positing that the richness of natural scenes is reflected in phenomenal consciousness but not in detail in the phenomenal judgments upon which reports about qualia are based. Conceptual awareness (including phenomenal judgments) arises from neural mechanisms that categorize objects, and also from mechanisms that conceptually characterize textural properties of pre-categorically segmented regions in the (...) visual field. Experimental evidence suggests that complex images instigate the generation of so-called ensemble phenomenal judgments. These involve concepts that categorize global attributes of segmented areas but carry no information pertaining to details. It is then argued that there are cogent reasons for believing that phenomenal percepts (i.e. qualia) arising from chromatically complex stimuli cohere in this ensemble sense with both the stimulus and with the resulting ensemble phenomenal judgments. Thus, spatially detailed retinal images are deemed to yield correspondingly detailed phenomenal experiences that are in turn conceptually apprehended via a relatively small number of ensemble phenomenal judgments. Lastly, it is suggested that the bridge locus for chromatically rich phenomenal experiences is most plausibly located early in the cortical visual pathway. (shrink)

As an artist I admittedly scrutinize all of the theories related to the arts closely. I do this for a number of reasons. The obvious one is that I have a deeply felt personal relationship with the subject matter. Less obvious is my experience in general. My early research was motivated by a desire to discover the historical circumstances that led to the difficulty in fitting visual art into the discussions I encountered. Generally, it seemed that the dominant framework (...) trivialized what I considered the most important aspects of the creative process. Over time I concluded that developing an interdisciplinary approach offered the best option for expanding views, although it is not an easy task. Establishing areas of commonality across a range of disciplines must somehow accommodate the ways in which each has developed a research agenda that seems to serve its core needs. In consciousness studies, for example, we have a field that relies heavily on scientific research and humanistic methodologies when building the philosophical models scholars use to structure theories. This methodology is not only removed from the nuts and bolts of art, it is also easily manipulated in discourse on art due to the ease with which we can fit aspects of art into the philosophical framework. Clearly this approach fits nicely with philosophically defined concepts such as meaning, emotion, and other elusive modes. In addition, using the well-honed categories aids in bracketing themes such as metaphor, interpretation, subjectivity, language and history. Nonetheless, in reading through the studies, I repeatedly conclude that the voices of practitioners need to be included to a greater degree. (shrink)

We prove consistent, assuming there is a supercompact cardinal, that there is a singular strong limit cardinalμ, of cofinalityω, such that everyμ+-chromatic graphXonμ+has an edge colouringcofXintoμcolours for which every vertex colouringgofXinto at mostμmany colours has ag-colour class on whichctakes every value.The paper also contains some generalisations of the above statement in whichμ+is replaced by other cardinals >μ.

We prove consistent, assuming there is a supercompact cardinal, that there is a singular strong limit cardinal $\mu$ , of cofinality $\omega$ , such that every $\mu^{+}$ -chromatic graph X on $\mu^{+}$ has an edge colouring c of X into $\mu$ colours for which every vertex colouring g of X into at most $\mu$ many colours has a g-colour class on which c takes every value. The paper also contains some generalisations of the above statement in which $\mu^{+}$ is (...) replaced by other cardinals < $\mu$. (shrink)

We show that there exist 2 ℵ 0 equational classes of Boolean algebras with operators that are not generated by the complex algebras of any first-order definable class of relational structures. Using a variant of this construction, we resolve a long-standing question of Fine, by exhibiting a bimodal logic that is valid in its canonical frames, but is not sound and complete for any first-order definable class of Kripke frames (a monomodal example can then be obtained using simulation results of (...) Thomason). The constructions use the result of Erd $\H{o}$ s that there are finite graphs with arbitrarily large chromatic number and girth. (shrink)

It is shown that if ZF is consistent, then so is ZFC + GCH + "There is a graph with cardinality ℵ 2 and chromatic number ℵ 2 such that every subgraph of cardinality ≤ ℵ 1 has chromatic number ≤ ℵ 0 ". This partially answers a question of Erdos and Hajnal.

A cylindric algebra atom structure is said to be strongly representable if all atomic cylindric algebras with that atom structure are representable. This is equivalent to saying that the full complex algebra of the atom structure is a representable cylindric algebra. We show that for any finite n >3, the class of all strongly representable n-dimensional cylindric algebra atom structures is not closed under ultraproducts and is therefore not elementary. Our proof is based on the following construction. From an arbitrary (...) undirected, loop-free graph Γ, we construct an n-dimensional atom structure E(Γ), and prove, for infinite Γ, that E(Γ) is a strongly representable cylindric algebra atom structure if and only if the chromatic number of Γ is infinite. A construction of Erdős shows that there are graphs (k < ω)with infinite chromatic number, but having a non-principal ultraproduct $\prod _D\Gamma _k $ whose chromatic number is just two. It follows that $E(\Gamma _k )$ is strongly representable (each k < ω) but $\Pi _D E(\Gamma _k)$ is not. (shrink)

It is consistent relative to an inaccessible cardinal that ZF+DC holds, and the hypergraph of isosceles triangles on $\mathbb {R}^2$ has countable chromatic number while the hypergraph of isosceles triangles on $\mathbb {R}^3$ has uncountable chromatic number.

We give classical proofs, strengthenings, and generalizations of Lecomte’s characterizations of analytic ω-dimensional hypergraphs with countable Borel chromatic number.

We explore a general method based on trees of elementary submodels in order to present highly simplified proofs to numerous results in infinite combinatorics. While countable elementary submodels have been employed in such settings already, we significantly broaden this framework by developing the corresponding technique for countably closed models of size continuum. The applications range from various theorems on paradoxical decompositions of the plane, to coloring sparse set systems, results on graph chromatic number and constructions from point-set topology. (...) Our main purpose is to demonstrate the ease and wide applicability of this method in a form accessible to anyone with a basic background in set theory and logic. (shrink)

We sketch the ideas behind the use of chromatic numbers in establishing descriptive set-theoretic dichotomy theorems.

Reviewed Works:Edwin W. Miller, On a Property of Families of Sets.Ben Dushnik, E. W. Miller, Partially Ordered Sets.P. Erdos, Some Set-theoretical Properties of Graphs.G. Fodor, Proof of a Conjecture of P. Erdos.P. Erdos, R. Rado, A Partition Calculus in Set Theory.P. Erdos, R. Rado, Intersection Theorems for Systems of Sets.A. Hajnal, Some Results and Problems on Set Theory.P. Erdos, A. Hajnal, On a Property of Families of Sets.A. Hajnal, Proof of a Conjecture of S. Ruziewicz.P. Erdos, A. Hajnal, R. Rado, (...) Partition Relations for Cardinal Numbers.P. Erdos, A. Hajnal, On a Problem of B. Jonsson.P. Erdos, A. Hajnal, On Chromatic Number of Graphs and Set-Systems. (shrink)

We characterize Borel line graphs in terms of 10 forbidden induced subgraphs, namely the nine finite graphs from the classical result of Beineke together with a 10th infinite graph associated with the equivalence relation $\mathbb {E}_0$ on the Cantor space. As a corollary, we prove a partial converse to the Feldman–Moore theorem, which allows us to characterize all locally countable Borel line graphs in terms of their Borel chromatic numbers.

May the same graph admit two different chromatic numbers in two different universes? How about infinitely many different values? and can this be achieved without changing the cardinals structure? In this paper, it is proved that in Gödel’s constructible universe, for every uncountable cardinal $$\mu $$ below the first fixed-point of the $$\aleph $$ -function, there exists a graph $$\mathcal G_\mu $$ satisfying the following.

The relationship of Grundy and chromatic numbers of graphs in the context of Reverse Mathematics is investi-gated.

We generalize the [Formula: see text] dichotomy to doubly-indexed sequences of analytic digraphs. Under a mild definability assumption, we use this generalization to characterize the family of Borel actions of tsi Polish groups on Polish spaces that can be decomposed into countably-many Borel actions admitting complete Borel sets that are lacunary with respect to an open neighborhood of the identity. We also show that if the group in question is non-archimedean, then the inexistence of such a decomposition yields a special (...) kind of continuous embedding of [Formula: see text] into the corresponding orbit equivalence relation. (shrink)

We investigate the pair-splitting number $\germ{s}_{pair}$ which is a variation of splitting number, pair-reaping number $\germ{r}_{pair}$ which is a variation of reaping number and cardinal invariants of ideals on ω. We also study cardinal invariants of F σ ideals and their upper bounds and lower bounds. As an application, we answer a question of S. Solecki by showing that the ideal of finitely chromatic graphs is not locally Katětov-minimal among ideals not satisfying Fatou's lemma.

This chapter discusses the ordering of color percepts, and starts by presenting an overview of the critical issues surrounding the topic and by examining the relationship between stimuli and percepts. Certain types of variability were found by experimental psychology in the relationship between stimulus and response as a result of observation conditions. In the twentieth century, the view that the normal human color-vision system has a standard implementation and that all perceptual data are appropriately treated with normal statistical distribution methodology (...) became the standard paradigm. However, the idea that the large number of color percepts humans can experience must fit into some kind of ordering system is an old one, going as far back to Aristotle and his proposed color categories based on a scale of chromatic colors between white and black. The strengths and limitations of the more recently developed kinds of color order systems are also touched upon here. (shrink)

We are concerned here with recursive function theory analogs of certain problems in chromatic graph theory. The motivating question for our work is: Does there exist a recursive (countably infinite) planar graph with no recursive 4-coloring? We obtain the following results: There is a 3-colorable, recursive planar graph which, for all k, has no recursive k-coloring; every decidable graph of genus p ≥ 0 has a recursive 2(χ(p) - 1)-coloring, where χ(p) is the least number of colors which (...) will suffice to color any graph of genus p; for every k ≥ 3 there is a k-colorable, decidable graph with no recursive k-coloring, and if k = 3 or if k = 4 and the 4-color conjecture fails the graph is planar; there are degree preserving correspondences between k-colorings of graphs and paths through special types of trees which yield information about the degrees of unsolvability of k-colorings of graphs. (shrink)

Graph polynomials are graph parameters invariant under graph isomorphisms which take values in a polynomial ring with a fixed finite number of indeterminates. We study graph polynomials from a model theoretic point of view. In this paper we distinguish between the graph theoretic (semantic) and the algebraic (syntactic) meaning of graph polynomials. Graph polynomials appear in the literature either as generating functions, as generalized chromatic polynomials, or as polynomials derived via determinants of adjacency or Laplacian matrices. We show (...) that these forms are mutually incomparable, and propose a unified framework based on definability in Second Order Logic. We show that this comprises virtually all examples of graph polynomials with a fixed finite set of indeterminates. Finally we show that the location of zeros and stability of graph polynomials is not a semantic property. The paper emphasizes a model theoretic view. It gives a unified exposition of classical results in algebraic combinatorics together with new and some of our previously obtained results scattered in the graph theoretic literature. (shrink)

As past president of both the History of Science Society and the American Society of Church History, Ronald L. Numbers is uniquely qualified to assess the historical relations between science and Christianity. In this collection of his most recent essays, he moves beyond the clichés of conflict and harmony to explore the tangled web of historical interactions involving scientific and religious beliefs. In his lead essay he offers an unprecedented overview of the history of science and Christianity from the perspective (...) of the ordinary people who filled the pews of churchesor loitered around outside. Unlike the elite scientists and theologians on whom most historians have focused, these vulgar Christians cared little about the discoveries of Copernicus, Newton, and Einstein. Instead, they worried about the causes of the diseases and disasters that directly affected their lives and about scientists preposterous attempts to trace human ancestry back to apes. Far from dismissing opinion-makers in the pulpit, Numbers closely looks at two the most influential Protestant theologians in nineteenth-century America: Charles Hodge and William Henry Green. Hodge, after decades of struggling to harmonize Gods two revelationsin nature and in the Biblein the end famously described Darwinism as atheism. Green, on the basis of his careful biblical studies, concluded that Ussher's chronology was unreliable, thus opening the door for Christian anthropologists to accommodate the subsequent discovery of human antiquity. In Science without God Numbers traces the millennia-long history of so-called methodological naturalism, the commitment to explaining the natural world without appeals to the supernatural. By the early nineteenth century this practice was becoming the defining characteristic of science; in the late twentieth century it became the central point of attack in the audacious attempt of intelligent designers to redefine science. Numbers ends his reassessment by arguing that although science has markedly changed the world we live in, it has contributed less to secularizing it than many have claimed. Taken together, these accessible and authoritative essays form a perfect introduction to Christian attitudes towards science since the 17th century. (shrink)

Originally published in 1995, The Selected Works of George McCready Price is the seventh volume in the series, Creationism in Twentieth Century America, reissued in 2019. The volume brings together the original writings and pamphlets of George McCready Price, a leading creationist of the early antievolution crusade of the 1920s. McCready Price labelled himself the 'principal scientific authority of the Fundamentalists' and as a self-taught scientist he enjoyed more scientific repute amongst fundamentalists of the time. This interesting and unique collection (...) of original source material includes five of his writings between 1906 and 1924, challenging the new Darwinian theory of evolution and natural selection through his writings on the natural sciences. His literature covers the topics of evolution and biology and critiques biological arguments for evolution. He also wrote widely on geology offering his own alternative argument of 'flood geography' in opposition to the Darwinian theory concerning palaeontology and geology. This volume will be of interest to historians of natural history and the creationism movement, as well as scholars of religion and American history. (shrink)

Originally published in 1995, The Antievolution Works of Arthur I. Brown is the third volume in the series, Creationism in Twentieth Century America. The volume brings together original sources from the prominent surgeon and creationist Arthur I. Brown. Brown discredited evolution as it was contrary to the 'clear statements of scripture' which he believed infallible, stating evolution instead to be both a hoax and 'a weapon of Satan'. The works included focus on Brown's polemic through his early twentieth century writings. (...) The essays focus on his scientific investigations and provide a negative commentary upon Darwin's theory of evolution instead focusing on biblical explanations for evolution. As a scientist Brown's unique view of evolution from a creationist and scientific viewpoint provides a fascinating lens through which to view the historical debates surrounding evolution and provides a unique insight into how Darwinian theory affected both the scientific and religious communities. This book will be of interest to natural historians, and theologians as well as academics of philosophy and history. (shrink)