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HTTP/1.1 200 OK Date: Sat, 27 Dec 2025 16:41:51 GMT Content-Type: text/html; charset=utf-8 Transfer-Encoding: chunked Connection: keep-alive Vary: Accept-Encoding Cache-Control: no-store, max-age=0 Pragma: no-cache Expires: 0 X-Clacks-Overhead: GNU Terry Pratchett Content-Encoding: gzip Server: PythonAnywhere Erdős Problems

Problems have always been an essential part of my mathematical life. A well-chosen problem can isolate an essential difficulty in a particular area, serving as a benchmark against which progress in this area can be measured. It might be like a 'marshmallow', serving as a tasty tidbit supplying a few moments of fleeting enjoyment. Or it might be like an 'acorn', requiring deep and subtle new insights from which a mighty oak can develop. […]

In this note I would like to describe a variety of my problems which I would classify as my favorites. Of course, I can't guarantee that they are all 'acorns', but because many have thwarted the efforts of the best mathematicians for many decades (and have often acquired a cash reward for their solutions), it may indicate that new ideas will be needed, which can, in turn, lead to more general results, and naturally, to further new problems.

In this way, the cycle of life in mathematics continues forever.

Paul Erdős, Some of my favorite problems and results, 1997

There are 1111 problems in the database of which 434 (39%) have been solved.

RECENT PAPERS

967 On a problem of Erdős and Ingham - Yip arXiv (18/12/25)
1045 An improved lower bound to Erdos' problem concerning products of distances for fixed diameter - Sothanaphan arXiv (16/12/25)
114 The maximal length of the Erdős-Herzog-Piranian lemniscate length in high degree - Tao arXiv (13/12/25)
69 248 946 Quantitative correlations and some problems on prime factors of consecutive integers - Tao and Teräväinen arXiv (02/12/25)
1102 1103 Growth rates of sequences governed by the squarefree properties of its translates - van Doorn and Tao arXiv (02/12/25)
845 Smooth sums with small spacings - van Doorn and Everts arXiv (06/11/25)
707 Forbidden Sidon subsets of perfect difference sets, featuring a human-assisted proof - Alexeev and Mixon arXiv (22/10/25)
165 Improving $R(3,k)$ in just two bites - Hefty, Horn, King, and Pfender arXiv (22/10/25)
775 On cliques in hypergraphs - Gao arXiv (17/10/25)
355 Lacunary sequences whose reciprocal sums represent all rationals in an interval - van Doorn and Kovač arXiv (30/09/25)
565 An exponential upper bound for induced Ramsey numbers - Aragão, Campos, Dahia, Filipe, and Marciano arXiv (29/09/25)
511 Many lemniscates with large diameter - Huang arXiv (16/09/25)
320 A lower bound for the number of Egyptian fractions - Bettin, Grenié, Molteni, and Sanna arXiv (15/09/25)
547 Ramsey numbers of trees - Montgomery, Pavez-Signé, and Yan arXiv (08/09/25)
510 Polynomial bounds for the Chowla Cosine Problem - Bedert arXiv (08/09/25)
510 From small eigenvalues to large cuts, and Chowla's cosine problem - Jin, Milojeviç, Tomon, and Zhang arXiv (04/09/25)
169 Integer sets of large harmonic sum which avoid long arithmetic progressions - Walker arXiv (04/09/25)
376 Integers with small digits in multiple bases - Bloom and Croot arXiv (03/09/25)
527 Convergent points for random power series on the unit circle - Michelen and Sawhney arXiv (02/09/25)
944 Critical edge sets in vertex-critical graphs - Skottova and Steiner arXiv (12/08/25)
682 Rough numbers between consecutive primes - Gafni and Tao arXiv (11/08/25)
755 The number of regular simplices in higher dimensions - Clemen, Dumitrescu, and Liu arXiv (26/07/25)
844 The independence and clique cover numbers of the squarefree graph - Alexeev, Mixon, and Sawin arXiv (02/07/25)
609 Short monochromatic odd cycles - Janzer and Yip arXiv (17/06/25)
893 On the number of divisors of Mersenne numbers - Kovač and Luca arXiv (10/06/25)
391 Decomposing a factorial into large factors - Alexeev, Conway, Rosenfeld, Sutherland, Tao, Uhr, and Ventullo arXiv (03/06/25)
165 A new lower bound for the Ramsey numbers $R(3,k)$ - Campos, Jenssen, Michelen, and Sahasrabudhe arXiv (19/05/25)
132 756 958 959 On multiplicities of interpoint distances - Clemen, Dumitrescu, and Liu arXiv (09/05/25)
964 Ratios of consecutive values of the divisor function - Eberhard arXiv (27/05/25)
270 On the irrationality of certain super-polynomially decaying series - Crmarić and Kovač arXiv (25/04/25)
411 On an iterated arithmetic function problem of Erdős and Graham - Steinerberger arXiv (14/04/25)
443 An elementary question of Erdős and Graham - Hegyvári arXiv (31/03/25)
816 A problem of Erdős and Hajnal on paths with equal-degree endpoints - Chen and Ma arXiv (25/03/25)
315 On a conjecture of Erdős and Graham about the Sylvester's sequence - Li and Tang arXiv (21/03/25)
244 On a Romanoff type problem of Erdős and Kalmár - Ding arXiv (17/03/25)
648 On Erdős problem #648 - Cambie arXiv (13/03/25)
315 Asymptotic analysis of infinite decompositions of a unit fraction into unit fractions - Kamio arXiv (04/03/25)
622 Cyclic subsets in regular Dirac graphs - Draganic, Keevash, and Muyesser arXiv (03/03/25)
143 Erdős's integer dilation approximation problem and GCD graphs - Koukoulopoulos, Lamzouri, and Lichtman arXiv (14/02/25)
792 Large sum-free subsets of sets of integers via $L^1$-estimates for trigonometric sums - Bedert arXiv (13/02/25)
612 Sharp results for the Erdős, Pach, Pollack and Tuza problem - Cambie and Jooken arXiv (13/02/25)
204 A Question of Erdős and Graham on Covering Systems - Adenwalla arXiv (25/01/25)
84 Improved bound on the number of cycle sets - Nenadov arXiv (20/01/25)
690 692 Resolution of Erdős' problems about unimodularity - Cambie arXiv (20/01/25)
787 The Erdős-Moser sum-free set problem via improved bounds for $k$-configurations - Beker arXiv (20/01/25)
795 Sharp Bounds for Sets with Distinct Subset Products - Raghavan arXiv (06/01/25)
353 Isosceles trapezoids of unit area with vertices in sets of infinite planar measure - Koizumi arXiv (03/01/25)
353 Polygons of unit area with vertices in sets of infinite planar measure - Kovač and Predojević arXiv (16/12/24)
120 Some recent progress on the Erdős similarity conjecture - Jung, Lai, and Mooroogen arXiv (15/12/24)
609 Monochromatic odd cycles in edge-coloured complete graphs - Girão and Hunter arXiv (10/12/24)
91 503 A note on Erdős's mysterious remark - Kovács arXiv (09/12/24)
264 265 266 268 On several irrationality problems for Ahmes series - Kovač and Tao arXiv (27/11/24)
106 A note on the Erdős conjecture about square packing - Baek, Koizumi, and Ueoro arXiv (18/11/24)
290 On the non-monotonicity of the denominator of generalized harmonic sums - van Doorn arXiv (05/11/24)
11 On the Sum of Squarefree Integers and a Power of Two - Hercher arXiv (04/11/24)
778 On a clique-building game of Erdős - Malekshahian and Spiro arXiv (25/10/24)
77 Upper bounds for multicolour Ramsey numbers - Balister, Bollobás, Campos, Griffiths, Hurley, Morris, Sahasrabudhe, and Tiba arXiv (22/10/24)
426 Unique subgraphs are rare - Bradač and Christoph arXiv (21/10/24)
186 Sharp bound for the Erdős-Straus non-averaging set problem - Pham and Zakharov arXiv (18/10/24)
641 Chromatic number and regular subgraphs - Janzer, Steiner, and Sudakov arXiv (03/10/24)
69 The irrationality of a prime factor series under a prime tuples conjecture - Pratt arXiv (23/09/24)
475 Graham's rearrangement conjecture beyond the rectification barrier - Bedert and Kravitz arXiv (11/09/24)
79 Infinitely many minimally Ramsey size-linear graphs - Wigderson arXiv (09/09/24)
518 A proof of a conjecture of Erdős and Gyárfás on monochromatic path covers - Pokrovskiy, Versteegen, and Williams arXiv (05/09/24)
150 On a question of Erdős and Nešetřil about minimal cuts in a graph - Bradač arXiv (04/09/24)
135 Planar point sets with forbidden 4-point patterns and few distinct distances - Tao arXiv (02/09/24)
395 The Reverse Littlewood-Offord problem of Erdős - He, Juškevičius, Narayanan, and Spiro arXiv (28/08/24)
625 On a question of Erdős and Gimbel on the cochromatic number - Heckel arXiv (27/08/24)
625 762 On the difference between the chromatic and cochromatic number - Steiner arXiv (05/08/24)
77 Optimizing the CGMS upper bound on Ramsey numbers - Gupta, Ndiaye, Norin, and Wei arXiv (26/07/24)
442 Dense sets of natural numbers with unusually large least common multiples - Tao arXiv (08/07/24)
756 A problem of Erdős about rich distances - Bhowmick arXiv (02/07/24)
475 Rearranging small sets for distinct partial sums - Kravitz arXiv (01/07/24)
206 On eventually greedy best underapproximations by Egyptian fractions - Kovač arXiv (12/06/24)
314 On differences of two harmonic numbers - Lim and Steinerberger arXiv (31/05/24)
429 Sparse Admissible Sets and a Problem of Erdős and Graham - Weisenberg arXiv (20/05/24)
121 On product representations of squares - Tao arXiv (19/05/24)
29 An explicit economical additive basis - Jain, Pham, Sawhney, and Zakharov arXiv (15/05/24)
268 On the set of points represented by harmonic subseries - Kovać arXiv (13/05/24)
297 A question of Erdős and Graham on Egyptian fractions - Conlon, Fox, He, Mubayi, Pham, Suk, and Verstraëte arXiv (24/04/24)
585 Edge-disjoint cycles with the same vertex set - Chakraborti, Janzer, Methuku, and Montgomery arXiv (10/04/24)
47 294 297 300 305 310 On further questions regarding unit fractions - Liu and Sawhney arXiv (10/04/24)
297 On a problem involving unit fractions - Steinerberger arXiv (25/03/24)
3 Improved Bounds for Szemerédi's Theorem - Leng, Sah, and Sawhney arXiv (28/02/24)
208 Squarefree numbers in short intervals - Pandey arXiv (25/01/24)
213 On integer distance sets - Greenfeld, Iliopoulou, and Peluse arXiv (19/01/24)
549 On the Ramsey number of the double star - Dubó and Stein arXiv (02/01/24)

OPEN -> SOLVED

(Some are new solutions, some are old solutions new to this site. The date is when the site was updated with the change of status.)

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