• The Cameron–Erdős conjecture on sum-free sets of integers, proved by Ben Green.
• The Erdős–Burr conjecture
  Erdos–Burr conjecture
  In mathematics, the Erdős–Burr conjecture is an unsolved problem concerning the Ramsey number of sparse graphs. The conjecture is named after Paul Erdős and Stefan Burr, and is one of many conjectures named after Erdős; it states that the Ramsey number of graphs in any sparse family of graphs...
  
   on Ramsey numbers of graphs.
• The Erdős–Faber–Lovász conjecture
  Erdos–Faber–Lovász conjecture
  In graph theory, the Erdős–Faber–Lovász conjecture is an unsolved problem about graph coloring, named after Paul Erdős, Vance Faber, and László Lovász, who formulated it in 1972...
  
   on coloring unions of cliques.
• The Erdős–Graham conjecture in combinatorial number theory on monochromatic Egyptian fraction representations of unity.
• The Erdős–Gyárfás conjecture
  Erdos–Gyárfás conjecture
  In graph theory, the unproven Erdős–Gyárfás conjecture, made in 1995 by the prolific mathematician Paul Erdős and his collaborator András Gyárfás, states that any graph with minimum degree 3 contains a simple cycle whose length is a power of two...
  
   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. [Ramsey-type theorems, Discrete Applied Mathematics 25 (1989) 37-52]
• The Erdős–Heilbronn conjecture in combinatorial number theory on the number of sums of two sets of residues modulo a prime, proved by J.A. Dias da Silva and Y.O. Hamidoune in 1994.
• The Erdős–Lovász conjecture on weak/strong delta-systems (http://www.math-inst.hu/~p_erdos/1975-32.pdf, p. 406), proved by Michel Deza
  Michel Deza
  Michel Marie Deza is a Soviet and French mathematician, specializing in combinatorics, discrete geometry and graph theory...
  
  .
• The Erdős–Mollin–Walsh conjecture
  Powerful number
  A powerful number is a positive integer m such that for every prime number p dividing m, p2 also divides m. Equivalently, a powerful number is the product of a square and a cube, that is, a number m of the form m = a2b3, where a and b are positive integers. Powerful numbers are also known as...
  
   on consecutive triples of powerful numbers.
• The Erdős–Menger conjecture on disjoint paths in infinite graphs. (solved by Ron Aharoni
  Ron Aharoni
  Ron Aharoni is an Israeli mathematician, working in finite and infinite combinatorics. Aharoni is a professor at the Technion – Israel Institute of Technology, where he received his Ph.D. in mathematics in 1979. With Nash-Williams and Shelah he generalized the marriage theorem by obtaining the...
  
   and Eli Berger])
• The Erdős–Selfridge conjecture that a covering set
  Covering set
  In mathematics, a covering set for a sequence of integers refers to a set of prime numbers such that every term in the sequence is divisible by at least one member of the set...
  
   contains at least one odd member.
• The Erdős–Stewart conjecture on the Diophantine equation
  Diophantine equation
  In mathematics, a Diophantine equation is an indeterminate polynomial equation that allows the variables to be integers only. Diophantine problems have fewer equations than unknown variables and involve finding integers that work correctly for all equations...
  
   n! + 1 = p_k^a p_k+1^b (solved by Luca)
• The Erdős–Straus conjecture
  Erdos–Straus conjecture
  In number theory, the Erdős–Straus conjecture states that for all integers n ≥ 2, the rational number 4/n can be expressed as the sum of three unit fractions. Paul Erdős and Ernst G...
  
   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–Woods conjecture
  Erdos–Woods number
  In number theory, an Erdős–Woods number is a positive integer that has the following property:Consider a sequence of consecutive positive integers [a, a+1, \dots, a+k]. The number k is an Erdős–Woods number if there exists such a sequence, beginning with some number a, in which each of the elements...
  
   on numbers determined by the set of prime divisors of the following k numbers.
• 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
  Erdős–Turán conjecture on additive bases
  The Erdős–Turán conjecture is an old unsolved problem in additive number theory posed by Paul Erdős and Pál Turán in 1941.-History:...
  
   of natural numbers.
• A conjecture on quickly growing integer sequences with rational reciprocal series
  Sylvester's sequence
  In number theory, Sylvester's sequence is an integer sequence in which each member of the sequence is the product of the previous members, plus one. The first few terms of the sequence are:...
  
  .
• A conjecture on equitable coloring
  Equitable coloring
  In graph theory, an area of mathematics, an equitable coloring is an assignment of colors to the vertices of an undirected graph, in such a way that*No two adjacent vertices have the same color, and...
  
  s proven in 1970 by András Hajnal
  András Hajnal
  András Hajnal is an emeritus professor of mathematics at Rutgers University and a member of the Hungarian Academy of Sciences known for his work in set theory and combinatorics.-Biography:Hajnal was born on 13 May 1931, in Hungary....
  
   and Endre Szemerédi
  Endre Szemerédi
  Endre Szemerédi is a Hungarian mathematician, working in the field of combinatorics and theoretical computer science. He is the State of New Jersey Professor of computer science at Rutgers University since 1986...
  
   and now known as the Hajnal–Szemerédi theorem.
• A conjecture with Norman Oler on circle packing in an equilateral triangle
  Circle packing in an equilateral triangle
  Circle packing in an equilateral triangle is a packing problem in discrete mathematics where the objective is to pack n unit circles into the smallest possible equilateral triangle...
  
   with a number of circles one less than a triangular number
  Triangular number
  A triangular number or triangle number numbers the objects that can form an equilateral triangle, as in the diagram on the right. The nth triangle number is the number of dots in a triangle with n dots on a side; it is the sum of the n natural numbers from 1 to n...
  
  .

External links

• Fan Chung, "Open problems of Paul Erdős in graph theory"