In set theory, a Kurepa tree is a tree (T, <) of height ω1, each of whose levels is countable, and has at least ℵ2 many branches. This concept was introduced by Kurepa (1935). The existence of a Kurepa tree (known as the Kurepa hypothesis, though Kurepa originally conjectured that this was false) is consistent with the axioms of ZFC: Solovay showed in unpublished work that there are Kurepa trees in Gödel's constructible universe (Jech 1971). More precisely, the existence of Kurepa trees follows from the diamond plus principle, which holds in the constructible universe. On the other hand, Silver (1971) showed that if a strongly inaccessible cardinal is Lévy collapsed to ω2 then, in the resulting model, there are no Kurepa trees. The existence of an inaccessible cardinal is in fact equiconsistent with the failure of the Kurepa hypothesis, because if the Kurepa hypothesis is false then the cardinal ω2 is inaccessible in the constructible universe.
A Kurepa tree with fewer than 2ℵ1 branches is known as a Jech–Kunen tree.
More generally if κ is an infinite cardinal, then a κ-Kurepa tree is a tree of height κ with more than κ branches but at most |α| elements of each infinite level α<κ, and the Kurepa hypothesis for κ is the statement that there is a κ-Kurepa tree. Sometimes the tree is also assumed to be binary. The existence of a binary κ-Kurepa tree is equivalent to the existence of a Kurepa family: a set of more than κ subsets of κ such that their intersections with any infinite ordinal α<κ form a set of cardinality at most α. The Kurepa hypothesis is false if κ is an ineffable cardinal, and conversely Jensen showed that in the constructible universe for any uncountable regular cardinal κ there is a κ-Kurepa tree unless κ is ineffable.
This concept was introduced by Kurepa (1935). The existence of a Kurepatree (known as the Kurepa hypothesis, though Kurepa originally conjectured that this...
Yugoslav mathematician. Milan Kurepa (1933–2000), Serbian atomic physicist. Kurepatree, a mathematical object from set theory Srpska Dijaspora, Poreklo Srpskih...
-Aronszajn trees). They are named for Nachman Aronszajn, who constructed an Aronszajn tree in 1934; his construction was described by Kurepa (1935). A...
tree is an Aronszajn tree. The existence of a Suslin tree is independent of ZFC, and is equivalent to the existence of a Suslin line (shown by Kurepa...
imply the existence of a Kurepatree, but the stronger ◊+ principle implies both the ◊ principle and the existence of a Kurepatree. Akemann & Weaver (2004)...
from Chang's conjecture (Magidor) or even from the nonexistence of a Kurepatree (Shelah). A weaker, still unsolved conjecture states that if |A|<min(A)...
with discovering Silver indiscernibles and generalizing the notion of a Kurepatree (called Silver's Principle). He discovered 0# ("zero sharp") in his 1966...
that CH does not imply the existence of a Suslin line. Existence of Kurepatrees is independent of ZFC, assuming consistency of an inaccessible cardinal...
mathematician and physicist, born in Osijek Đuro Kurepa (1907–1993) - mathematician, best known for the Kurepatree Gojko Nikoliš (1911–1995), doctor, general...
Notable Serb mathematicians include Mihailo Petrović, Jovan Karamata and Đuro Kurepa. Mihailo Petrović is known for having contributed significantly to differential...
statements can be calibrated by large cardinals. For example: the negation of Kurepa's hypothesis is equiconsistent with the existence of an inaccessible cardinal...
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