List of conjectures by Paul Erdős
The prolific mathematician Paul Erdős and his various collaborators made many famous mathematical conjectures, over a wide field of subjects, and in many cases Erdős offered monetary rewards for solving them.
Unsolved
- The Erdős–Faber–Lovász conjecture on coloring unions of cliques.
- The Erdős–Gyárfás conjecture on cycles with lengths equal to a power of two in graphs with minimum degree 3.
- The Erdős–Hajnal conjecture that in a family of graphs defined by an excluded induced subgraph, every graph has either a large clique or a large independent set.[1]
- The Erdős–Mollin–Walsh conjecture on consecutive triples of powerful numbers.
- The Erdős–Selfridge conjecture that a covering system with distinct moduli contains at least one even modulus.
- The Erdős–Straus conjecture on the Diophantine equation 4/n = 1/x + 1/y + 1/z.
- The Erdős conjecture on arithmetic progressions in sequences with divergent sums of reciprocals.
- The Erdős–Szekeres conjecture on the number of points needed to ensure that a point set contains a large convex polygon.
- The Erdős–Turán conjecture on additive bases of natural numbers.
- A conjecture on quickly growing integer sequences with rational reciprocal series.
- A conjecture with Norman Oler on circle packing in an equilateral triangle with a number of circles one less than a triangular number.
- The minimum overlap problem to estimate the limit of M(n).
- A conjecture on whether the ternary expansion of contains at least one digit 2, for .[2]
Solved
- The Erdős sumset conjecture on sets, proven by Joel Moreira, Florian Karl Richter, Donald Robertson in 2018. The proof has appeared in "Annals of Mathematics" in March 2019.[3]
- The Burr–Erdős conjecture on Ramsey numbers of graphs, proved by Choongbum Lee in 2015.
- A conjecture on equitable colorings proven in 1970 by András Hajnal and Endre Szemerédi and now known as the Hajnal–Szemerédi theorem.[4]
- A conjecture that would have strengthened the Furstenberg–Sárközy theorem to state that the number of elements in a square-difference-free set of positive integers could only exceed the square root of its largest value by a polylogarithmic factor, disproved by András Sárközy in 1978.[5]
- The Erdős–Lovász conjecture on weak/strong delta-systems, proved by Michel Deza in 1974.[6]
- The Erdős–Heilbronn conjecture in combinatorial number theory on the number of sums of two sets of residues modulo a prime, proved by Dias da Silva and Hamidoune in 1994.[7]
- The Erdős–Graham conjecture in combinatorial number theory on monochromatic Egyptian fraction representations of unity, proved by Ernie Croot in 2000.[8]
- The Erdős–Stewart conjecture on the Diophantine equation n! + 1 = pka pk+1b, solved by Florian Luca in 2001.[9]
- The Cameron–Erdős conjecture on sum-free sets of integers, proved by Ben Green and Alexander Sapozhenko in 2003–2004.[10]
- The Erdős–Menger conjecture on disjoint paths in infinite graphs, proved by Ron Aharoni and Eli Berger in 2009.[11]
- The Erdős distinct distances problem. The correct exponent was proved in 2010 by Larry Guth and Nets Katz, but the correct power of log n is still open.[12]
- Erdős-Rankin conjecture on prime gaps, proved by Ford, Green, Konyagin, and Tao in 2014
- Erdős discrepancy problem on partial sums of ±1-sequences.
- Erdős squarefree conjecture that central binomial coefficients C(2n, n) are never squarefree for n > 4 was proved in 1996.
References
- Erdős, P.; Hajnal, A. (1989), "Ramsey-type theorems", Combinatorics and complexity (Chicago, IL, 1987), Discrete Applied Mathematics, 25 (1–2): 37–52, doi:10.1016/0166-218X(89)90045-0, MR 1031262.
- Lagarias, Jeffrey C. (2009), "Ternary expansions of powers of 2", Journal of the London Mathematical Society, Second Series, 79 (3): 562–588, arXiv:math/0512006, doi:10.1112/jlms/jdn080, MR 2506687
- Moreira, J.; Richter, F. K.; Robertson, D. (2019), "A proof of a sumset conjecture of Erdős", Annals of Mathematics, 189 (2): 605–652, arXiv:1803.00498, doi:10.4007/annals.2019.189.2.4, MR 3919363, Zbl 1407.05236.
- Hajnal, A.; Szemerédi, E. (1970), "Proof of a conjecture of P. Erdős", Combinatorial theory and its applications, II (Proc. Colloq., Balatonfüred, 1969), North-Holland, pp. 601–623, MR 0297607.
- Sárközy, A. (1978), "On difference sets of sequences of integers. II", Annales Universitatis Scientiarum Budapestinensis de Rolando Eötvös Nominatae, 21: 45–53 (1979), MR 0536201.
- Deza, M. (1974), "Solution d'un problème de Erdős-Lovász", Journal of Combinatorial Theory, Series B (in French), 16 (2): 166–167, doi:10.1016/0095-8956(74)90059-8, MR 0337635.
- da Silva, Dias; A., J.; Hamidoune, Y. O. (1994), "Cyclic spaces for Grassmann derivatives and additive theory", Bulletin of the London Mathematical Society, 26 (2): 140–146, doi:10.1112/blms/26.2.140.
- Croot, Ernest S., III (2000), Unit Fractions, Ph.D. thesis, University of Georgia, Athens. Croot, Ernest S., III (2003), "On a coloring conjecture about unit fractions", Annals of Mathematics, 157 (2): 545–556, arXiv:math.NT/0311421, Bibcode:2003math.....11421C, doi:10.4007/annals.2003.157.545.
- Luca, Florian (2001), "On a conjecture of Erdős and Stewart", Mathematics of Computation, 70 (234): 893–896, Bibcode:2001MaCom..70..893L, doi:10.1090/S0025-5718-00-01178-9, MR 1677411.
- Sapozhenko, A. A. (2003), "The Cameron-Erdős conjecture", Doklady Akademii Nauk, 393 (6): 749–752, MR 2088503. Green, Ben (2004), "The Cameron-Erdős conjecture", Bulletin of the London Mathematical Society, 36 (6): 769–778, arXiv:math.NT/0304058, doi:10.1112/S0024609304003650, MR 2083752.
- Aharoni, Ron; Berger, Eli (2009), "Menger's Theorem for infinite graphs", Inventiones Mathematicae, 176 (1): 1–62, arXiv:math/0509397, Bibcode:2009InMat.176....1A, doi:10.1007/s00222-008-0157-3.
- Guth, l.; Katz, N. H. (2010), On the Erdős distinct distance problem on the plane, arXiv:1011.4105, Bibcode:2010arXiv1011.4105G.
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