Great problems

20th century

1. 1935
   Happy ending problem (Erdős–Szekeres)Combinatorics and discrete mathematics, Geometry. How many points in general position in the plane are needed to guarantee a convex $n$-gon? Posed by Esther Klein for $n = 4$ in 1933—leading to her marriage to George Szekeres and Paul Erdős's nickname "the Happy Ending Problem"—the conjectured exact value $\mathrm{ES}(n) = 2^{n-2} + 1$ is proved only for $n \le 6$, though Andrew Suk established $\mathrm{ES}(n) = 2^{n + o(n)}$ in 2016.ErdősOpen
2. 1941
   Erdős–Turán conjecture on additive basesArithmetic and number theory, Combinatorics and discrete mathematics. If a set of natural numbers eventually covers all integers by pairwise sums (an asymptotic basis of order $2$), must some integers have arbitrarily many representations as a sum of two elements? Posed by Paul Erdős and Pál Turán in 1941, the conjecture $\limsup_{n \to \infty} r_A(n) = \infty$ remains open.ErdősOpen
3. 1946
   Erdős unit distance problemGeometry, Combinatorics and discrete mathematics. How many pairs among $n$ points in the plane can lie at distance $1$? Bounded above by $O(n^{4/3})$, while a 2026 number-theoretic construction disproved Erdős's $n^{1+o(1)}$ conjecture by proving $u(n) = \Omega(n^{1+c})$.ErdősOpen
4. 1948
   Erdős–Straus conjectureArithmetic and number theory. Formulated in 1948 by Paul Erdős and Ernst G. Straus, this Egyptian-fraction conjecture asks whether $4/n$ is always a sum of three reciprocals of positive integers. Because any decomposition for a divisor $d \mid n$ scales to a decomposition for $n$, it suffices to prove the statement when $n = p$ is prime. Simple algebraic identities immediately cover primes $p \not\equiv 1 \pmod{24}$, and larger modular covering systems eliminate almost all residue classes, enabling computer verification up to $n = 10^{17}$, yet a complete covering system of finitely many identities cannot exist.ErdősOpen
5. 1960
   Erdős–Rado sunflower conjectureCombinatorics and discrete mathematics. How large can a family of $w$-element sets be without containing $r$ sets whose pairwise intersections are all identical? Posed by Paul Erdős and Richard Rado in 1960 with the factorial bound $w! (r - 1)^w$, the problem saw a landmark breakthrough in 2019 by Alweiss, Lovett, Wu, and Zhang (refined to $(C r \log w)^w$), while the pure exponential bound $C_r^w$ remains open.ErdősOpen
6. 1973
   Erdős conjecture on arithmetic progressionsArithmetic and number theory, Combinatorics and discrete mathematics. If the sum of reciprocals $\sum_{n \in A} \frac{1}{n} = \infty$, must $A$ contain arbitrarily long arithmetic progressions? Proved for $k = 3$ by Thomas Bloom and Olof Sisask (2020), while all lengths $k \ge 4$ remain open.ErdősOpen
7. 1977
   Erdős–Hajnal conjectureCombinatorics and discrete mathematics. Whereas general $n$-vertex graphs need only contain a clique or independent set of logarithmic size $\Theta(\log n)$, Paul Erdős and András Hajnal conjectured in 1977 that forbidding any fixed induced subgraph $H$ forces a polynomial-size clique or independent set $n^{\delta_H}$.ErdősOpen