Well there are different levels of "correct".
Most of the bits of this answer have been given separately in other answers or comments but I'd like to submit a "complete" answer if I may.
The most correct would be to write the whole name the Hungarian way:
In Hungarian, surnames come first, given names come last. This is also the case in East Asia with Chinese, Japanese, and Korean at least.
Pál and Paul are the Hungarian and English versions respectively of the same given name popular across Christian Europe after Saint Paul the Apostle (Παῦλος in Biblical Greek, Paulus in Latin).
Pál is pronounced with a "long a" so might be written "phonetically" as "Pahl". For speakers of non-rhotic English varieties (don't pronounce "r" after a vowel) this rhymes with "Carl". Wikipedia gives the IPA pronunciation
English and Hungarian have different spelling "quirks". Hungarian s sounds most like English sh whereas the English s sound is spelled sz in Hungarian.
Hungarian "Umlauts" (better technical words are "diaeresis" and "trema") represent different vowel qualities, much as they do in German, Swedish, etc. Hungarian "Acute accents" indicate vowels that are pronounced longer. "Double acutes" are actually a combination of the other two, so long versions of vowels that have different sound to the letter without the diacritics.
Wikipedia gives the IPA for Erdős as
ɛ occurs in various varieties of English. It may sound closer to the "e" in "bet" or the "a" in "bat" if you speak a rhotic variety but for speakers of non-rhotic varieties the closest sound is probably that of "air".
ø Does not occur in English but is most similar to the "er" of non-rhotic varieties. If you speak French or German it's closer to "eu" in the former and "ö" in the latter.
English typewriters, keyboards, and older digital technology did not support any "accented" letters. Many European ones supported
ö but not
ö is seen as looking more like the less well-known
ő than the plain
o so was often used by people that knew a way to type or enter it.
All of this means there are many possible variations depending on how many of the above facts you know, combined with your personal feelings of what's correct in English when writing foreign names, modulated by how you've seen the name written before.
So you will see Pal, Pál, and Paul with Erdős, Erdös, and Erdos, in either order.
If you prefer to use popularity as an arbiter, Google Ngram Viewer only seems to cover Paul Erdos and Paul Erdös. The former is four to five times more common but declining, while the latter is gaining popularity.