What are Erdős problems?

Problems or conjectures posed (either individually or jointly with others) by the great mathematician Paul Erdős who posed many challenging questions for over six decades that have served as a benchmark of progress in many diverse fields of mathematics.

Who's behind this website?

This website was made by Thomas Bloom, a mathematician who likes to think about the problems Erdős posed. Technical assistance with setting up the code for the website was provided by ChatGPT and the logo was made by Midjourney. Since this website was launched, many people have helped with spotting typos and pointing to updated references, or suggesting new problems. These people are credited in the small print under each problem.

Where are the remarks below each problem from?

Some from Erdős himself, in the discussion surrounding the problem in a problem list he wrote. I have sometimes expanded this by including updated references and also other remarks. If you're interested in any particular problem, I encourage you to go and look at the original problem descriptions by Erdős (sources in the bottom left of the problem box), which often contain more information than I have chosen to display here, and to do a literature search - while I'm trying to make sure the site records the current state of the art, this is a work in progress!

Your statement of the problem looks different to what Erdős wrote. Why?

Often I found the way Erdős stated a problem to be hard to understand or unnecessarily verbose. I have sometimes taken the liberty of changing the statement into what is (in my opinion) the most elegant/easiest to parse version. If you think that I have accidentally changed the actual content of the problem, please let me know.

Is the database up to date (e.g. the open/solved status of each problem)?

No! That's the eventual goal, but for now I'm just adding the problems in manually from Erdős's problem lists, and it would take a lot longer to then go and do a literature search on what has been done on each problem since. There are also problems where I know that progress has been made but haven't gotten around to updating the database yet. I hope to update all problems eventually, but this will take a while. If you have an update on the status of any problem, please email me at erdosproblemsonline@gmail.com.

If you are interested in any problem I encourage you to go and search around in the first instance, to discover what has been done and if it has been solved. If you find any useful references that you think should be added here, please let me know.

I've found a mistake/would like to add a reference/more contextual remarks to a problem.

Thanks! The best way to do this (for now at least) is to send an email to erdosproblemsonline@gmail.com (including the change and the problem id).

Are all Erdős problems on this website?

Definitely not, adding these is a work in progress, and the hope is that this website will become more comprehensive over time. If you'd like to speed up this process, you can send in new Erdős problems yourself to erdosproblemsonline@gmail.com - it'll appear faster if you can include the source and a complete LaTex markup of the problem (with no macros!).

I also don't plan to include every open problem that Erdős ever posed; he was very prolific and wrote over 1500 papers, many of which had open problems, and wrote many letters. My (personal and subjective) aim is to include all of the **interesting** Erdős problems. For example, sufficient (but not necessary) conditions for a problem being included are Erdős including it on a standalone published list of problems, or being on record as offering a monetary value for its solution. A usually necessary (but not sufficient) condition is that the problem be reasonably understandable and interesting as a stand-alone statement (e.g. stating at the end of a long technical lemma in a long technical paper 'can this be improved?' does not count).

Why is there a monetary value attached to some problems?

Erdős often attached prizes to his problems. Often this is a reasonable measure of how interesting/difficult Erdős thought the problem was. He was not always consistent with the value of the prizes - where there is ambiguity I have chosen to show the highest prize value that Erdős is on record as offering for that problem.

Will you pay me the prize if I solve a problem?

No.

Will anyone?

As far as I know, not anymore.

I have a question not listed here.

Ask at erdosproblemsonline@gmail.com.