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declared open · formalizederdos:501
For every x∈Rx \in \mathbb{R} let Ax⊂RA_x \subset \mathbb{R} be a bounded set of Lebesgue outer measure <1< 1. Must there be an infinite independent set, that is an infinite X⊆RX \subseteq \mathbb{R} with x∉Ayx \notin A_y for all distinct x,y∈Xx, y \in X? Erdős and Hajnal proved that arbitrarily large finite independent sets exist. Hechler showed in 1972 that the answer is no under the continuum hypothesis, so any positive answer had to come from a model where CH fails, and Sungchul Lee later derived one from a real-valued measurable cardinal. The answer is that neither side is provable. Dropping Lee's large cardinal by transferring his argument to the extension of a model of CH by random reals gives a model where the answer is yes; Hechler's construction gives one where it is no. The question is independent of ZFC.
combinatoricsset theoryN/A4 sources
declared open · formalized · $500erdos:593
Erdos Problem #593 asks which finite triple systems occur in every uncountably chromatic triple system. The answer is exactly the class generated from private-vertex expansions of finite bipartite graphs by finite disjoint unions and one-point amalgamations. Problem #1177 on exact spectra is settled alongside it.
set theorygraph theoryhypergraphsN/A4 sources
declared solvederdos:603
Erdős problem 603
combinatoricsset theoryN/A3 sources
declared open · formalizederdos:623
Let XX be a set of cardinality ℵω\aleph_\omega and ff a function from the finite subsets of XX to XX such that f(A)∉Af(A)\not\in A for all AA. Must there exist an infinite independent Y⊆XY\subseteq X, i.e. with f(B)∉Yf(B)\not\in Y for all finite B⊂YB\subset Y? Claimed resolution: the positive assertion is equivalent to Koepke's free-subset property, hence independent of ZFC, with consistency strength exactly a measurable cardinal.
set theoryN/A4 sources
declared openerdos:1177
For a finite forbidden triple system GG, what exact uncountable chromatic cardinalities occur among GG-free triple systems, and how do those spectra interact? The revised manuscript answers the three exact-cardinal questions and claims a complete spectrum dichotomy.
set theorychromatic numberhypergraphsN/A3 sources