On transitive subrelations of binary relations

Abstract

The transitive closure of a binary relation R can be thought of as the best possible approximation of R “from above” by a transitive relation. We consider the question of approximating a relation from below by transitive relations. Our main result is that every thick relation (a relation whose complement contains no infinite chain) on a countable set has a transitive thick subrelation. This allows for a solution to a problem arising from previous work by the author and Alan Taylor. We also exhibit a thick relation on an uncountable set with no transitive thick subrelation.

Universality of the μ-predictor

“…The direction (i)⇒(ii) always holds, and (ii)⇒(i) is known to hold when X is countable or V is transitive [Har11]. Also, if (ii)⇒(i) holds for acyclic V , then it holds for all V (since intersecting V with a well-ordering of X makes V acyclic while preserving (ii)).…”

“…Currently, the only known technique for producing good predictors based on the µ-predictor for nontransitive visibility relations is to voluntarily coarsen the notion of indistinguishability to one that is more cooperative, without coarsening it too much. For example, given a nontransitive visibility relation V , we can often find a transitive T ⊆ V that is "close" to V in some sense, and use the µ-predictor with T as our notion of visibility; see [Har11] for details. In that same paper, an example is given for which that approach cannot be made to work; specifically, a nontransitive V is constructed that holds some promise for admitting a finite-error predictor, but for which no transitive subrelation admits a finite-error predictor.…”

Minimal predictors in hat problems

“…For some kinds of hat problems, the results in the transitive case carry over to the nontransitive case, but are just (apparently) harder to prove. For example, in the transitive case with two or more colors, a finite-error predictor exists iff there is no sequence x n of players such that x i cannot see x j for i ≤ j, and the proof is fairly succinct; in the nontransitive case, the same holds when there are countably many players (the uncountable case is open), but the proof is more elaborate [Har10]. So, it would be natural to speculate that the situation with minimal predictors is similar, and that Theorem 3.1 holds in the nontransitive case, but with a more complicated proof.…”