Here are a few conjectures proposed by Erdős that, despite some progress, remain unsolved...
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Analytic and probabilistic number theory was one of Paul Erdős's favorite subjects. Here is a small selection of his contributions in this field, picked here and there from topics that can still be presented accessibly.
The Erdős and Rényi random graph model is so famous in mathematics that it has become a common noun: in probability theory, people routinely speak of "an Erdős-Rényi" the way one would speak of a Brillat-Savarin in gastronomy or a Stradivarius in music.
The "Book" was a project of Erdős's to bring together the most beautiful known mathematical proofs, several of which had been found by Erdős himself. Although he died shortly before the project was completed, the work was nonetheless published, authored by Aigner and Ziegler. Its successive new editions keep the flame alive.
Singmaster's conjecture, the Polymath projects, the Green–Tao theorem…
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