How to find Erdős' treasure trove? The renowned mathematician, Paul Erdős, has published more than 1500 papers in various branches of mathematics including discrete mathematics, graph theory, number theory, mathematical analysis, approximation theory, set theory, and probability theory. A complete list of his published works is available in an archive organized by Jerrold Grossman. 
Besides the published papers, it is quite likely for a very active mathematician like Erdős to have a long list of unpublished/unfinished papers as well. Just like a treasure trove, these notes may contain incredibly valuable mathematical stuff such as conjectures, lemmas, proof ideas, etc., which can easily be expanded to full papers or at least, give rise to some interesting piece of research as a source of inspiration. They could be of some historical significance too.  

Question. Is there an archive of (even a portion of) Erdős' unpublished/unfinished works? I am particularly interested in those unpublished notes of him which are related to set theory and infinitary combinatorics.  


Update. Thanks to Ron Graham the question is now fully answered with the identification of the exact location of Erdős' mathematical diaries. Unfortunately, it seems these diaries aren't currently available for public (and even for close friends and colleagues of Erdős). See my below answer for further details.      
 A: Paul Erdős's notes on Egyptian fractions are with Ronald Graham, who has reproduced some of them in Paul Erdős and Egyptian Fractions. Graham mentions one unfinished manuscript in which "it is shown that any integer can be represented as a sum of reciprocals of distinct numbers which each have exactly three prime factors". This was only published in 2015. As this commentary aptly notes "Nearly 20 years after his death, the famed mathematician Paul Erdős keeps on publishing, thanks to the conjectures he left behind and the friends who strive to prove them."

The 2015 publication by Butler, Erdős, and Graham ends with this comment: "One of the authors believes that all rational numbers can be expressed in this form, another author has doubts that every rational number can be expressed in this form, and the third author, already having looked in The BOOK at the answer, remains silent on this issue."


An altogether different "treasure trove" is formed by the letters which Paul Erdős wrote throughout his life. The Archive for American Mathematics has digitised 435 letters from Erdős to Carl Pomerance, accessible here, and is solliciting further donations of correspondence. A commentary entitled New Gems in Old Letters says: 

Letters
  were a place in which Erdős put his mathematical thoughts in progress.
  Theorems are outlined, and new problems are suggested. Because he
  wrote so many letters, his collaborators sometimes never found time to
  follow up on all of these ideas; this leaves us with the rather
  shocking fact that many of Erdős's mathematical ideas are still sitting in
  drawers and filing cabinets of mathematicians around the world. Some of these
  may lead nowhere, but some are likely brilliant insights, still capable of having an
  impact on mathematics today.

A: Following my private communication with Ron Graham, he kindly shared all what he knew about Erdős' mathematical diaries with me. Here is part of our discussion which I am allowed to share on MathOverflow for the benefit of the public mathematical community. At Ron's request, I removed the name of one of Erdős' co-authors in the below passage. Emphases are mine. 
I would like to sincerely thank Ron for his attention to my question as well as his permission for publishing part of our private communication here on MathOverflow.

Besides the literally thousands of letters that Erdős wrote during his lifetime, the best source of his unpublished mathematical thoughts are contained in his mathematical diaries, which he meticulously kept for most of his life. In them, he would write what he was thinking about, who he was visiting, etc. There are 15 or so of them, kept in what looks like laboratory notebooks. Of course, they are all in Hungarian. When Erdős died, the diaries were given to his close colleague [name removed] (who) still has them but will not let anyone see them. Many of us have tried to change [name removed]'s mind but to no avail. 

