An introduction to infinite hat problems

Hardin, Christopher S.; Taylor, Alan D.

doi:10.1007/bf03038092

Cited by 13 publications

(19 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…However, it turns out to be somewhat of a disjoint union of what we have done here and the finite case in [4].…”

Section: Discussionmentioning

confidence: 75%

“…The question is whether or not they can devise a strategy ensuring that no matter how the hats are placed, only finitely many people will guess their own hat color incorrectly. Yuval Gabay and Michael O'Connor answered this in the affirmative (included in [4] with permission). Their proof easily generalizes to the context of ideals, where an ideal on a set X is a collection of subsets of X that contains all singletons and is closed under finite unions and subset formation.…”

Section: Consequencesmentioning

confidence: 94%

See 1 more Smart Citation

Limit-like predictability for discontinuous functions

Hardin

Taylor

2009

Proc. Amer. Math. Soc.

Self Cite

View full text Add to dashboard Cite

Abstract. Our starting point is the following question: To what extent is a function's value at a point x of a topological space determined by its values in an arbitrarily small (deleted) neighborhood of x? For continuous functions, the answer is typically "always" and the method of prediction of f (x) is just the limit operator. We generalize this to the case of an arbitrary function mapping a topological space to an arbitrary set. We show that the best one can ever hope to do is to predict correctly except on a scattered set. Moreover, we give a predictor whose error set, in T 0 spaces, is always scattered. Topological preliminariesAmong the topological spaces that we will be interested in are the ones arising from a partial ordering by either declaring a set to be open if it is closed upward in the ordering (the upward topology) or closed downward in the ordering (the downward topology). For example, the interval (−∞, 0] is open in the downward topology on the reals. These topologies are not typically T 1 , but they are always T 0 .We will be asserting that certain things happen except on a set that is "topologically small." On the real line R with the downward topology, we want these small sets to be the well-ordered subsets of R, and for the upward topology on an ordinal, we want these small sets to be the finite sets. The following well-known notions achieve both. Definition 1.1. A set S in a topological space X is weakly scattered if for every nonempty T ⊆ S there exists some x ∈ T and some neighborhood V of x such that V ∩ T is finite. We call such points weakly isolated points of T. The set S is scattered if the conclusion can be strengthened to V ∩ T = {x}, in which case these are called isolated points of T.What we are calling "weakly scattered" is called "separated" by Morgan [6]; he attributes the definition to Cantor. Every scattered set is weakly scattered, and it is straightforward to show that the two notions are equivalent in T 0 spaces. The concept of a weakly scattered set is a smallness notion in the sense that these sets are closed under the formation of subsets and finite unions.

show abstract

“…However, it turns out to be somewhat of a disjoint union of what we have done here and the finite case in [4].…”

Section: Discussionmentioning

confidence: 75%

Section: Consequencesmentioning

confidence: 94%

Limit-like predictability for discontinuous functions

Hardin

Taylor

2009

Proc. Amer. Math. Soc.

Self Cite

View full text Add to dashboard Cite

show abstract

“…Stated differently, for a fixed number of colors, we seek a characterization of those visibility graphs that yield a minimal predictor. Our first theorem answers this for the case of 2 colors and the case of n colors; the result appears as Theorem 1 in [HT08a], although most of it can be derived from results in [BHKL08]. But first we need a lemma that confirms an intuition about how many agents guess correctly on average.…”

Section: Minimal Predictorsmentioning

confidence: 88%

“…For the case of a visibility graph that is complete, there is a very satisfying result that we present below. It occurs as Theorem 2 in [BHKL08] and as Theorem 3 in [HT08a], although it was first proved for two colors by Peter Winkler [Win01] and later generalized to k colors by Uriel Feige [Fei04].…”

Section: Optimal Predictorsmentioning

confidence: 99%