02
Erdős unit-distance problem (upper bound) Open
Let u ( n ) u(n) u ( n ) denote the maximum number of unordered pairs at distance exactly 1 1 1 determined by n n n points in the Euclidean plane R 2 \mathbb{R}^2 R 2 . Erdős (1946) proved u ( n ) ≥ n 1 + c / log log n u(n) \geq n^{1 + c/\log\log n} u ( n ) ≥ n 1 + c / l o g l o g n via a n × n \sqrt{n} \times \sqrt{n} n × n integer grid construction (and the May 2026 disproof of his upper-bound conjecture shows u ( n ) ≥ n 1.014 … u(n) \geq n^{1.014\ldots} u ( n ) ≥ n 1.014 … for infinitely many n n n ). The best known upper bound is u ( n ) = O ( n 4 / 3 ) u(n) = O(n^{4/3}) u ( n ) = O ( n 4/3 ) . \textbf{Task:} Improve the upper bound exponent below 4 / 3 4/3 4/3 , or otherwise advance the asymptotic understanding of u ( n ) u(n) u ( n ) . erdos-unit-distance-upper-bound
attack → 03
Erdős-Straus conjecture Open
Prove that for every integer n ≥ 2 n \geq 2 n ≥ 2 , the equation 4 n = 1 a + 1 b + 1 c \dfrac{4}{n} = \dfrac{1}{a} + \dfrac{1}{b} + \dfrac{1}{c} n 4 = a 1 + b 1 + c 1 has a solution in positive integers a , b , c a, b, c a , b , c . 04
Erdős discrepancy problem — quantitative version Open
Tao (2015) proved Erdős's discrepancy conjecture: for every sequence f : N → { − 1 , + 1 } f : \mathbb{N} \to \{-1, +1\} f : N → { − 1 , + 1 } , sup N , d ∈ N ∣ ∑ j = 1 N f ( j d ) ∣ = ∞ \sup_{N, d \in \mathbb{N}} \left| \sum_{j=1}^{N} f(jd) \right| = \infty sup N , d ∈ N ∑ j = 1 N f ( j d ) = ∞ . The proof is non-effective. Define D ( N ) : = min f max n ≤ N , d ≥ 1 ∣ ∑ j = 1 n f ( j d ) ∣ D(N) := \min_{f} \max_{n \leq N,\, d \geq 1} \left| \sum_{j=1}^{n} f(jd) \right| D ( N ) := min f max n ≤ N , d ≥ 1 ∑ j = 1 n f ( j d ) where the minimum runs over all ± 1 \pm 1 ± 1 sequences. \textbf{Task:} Establish an explicit quantitative growth rate for D ( N ) → ∞ D(N) \to \infty D ( N ) → ∞ , i.e., an unconditional lower bound of the form D ( N ) ≥ g ( N ) D(N) \geq g(N) D ( N ) ≥ g ( N ) for an explicit unbounded function g g g . erdos-discrepancy-quantitative
attack → 06
Goldbach's binary conjecture Very hard
Prove that every even integer n ≥ 4 n \geq 4 n ≥ 4 can be written as a sum of two (not necessarily distinct) primes. 08
Hadwiger-Nelson problem (chromatic number of the plane) Open
Determine χ ( R 2 ) \chi(\mathbb{R}^2) χ ( R 2 ) , the smallest number of colors required to color the points of the Euclidean plane such that no two points at distance exactly 1 1 1 receive the same color. It is known that χ ( R 2 ) ∈ { 5 , 6 , 7 } \chi(\mathbb{R}^2) \in \{5, 6, 7\} χ ( R 2 ) ∈ { 5 , 6 , 7 } ; the task is to determine the exact value, or to strictly improve either the lower bound (currently 5 5 5 ) or the upper bound (currently 7 7 7 ). 10
Lonely runner conjecture Open
Suppose k ≥ 2 k \geq 2 k ≥ 2 runners run forever on a unit-circumference circular track at distinct constant nonzero speeds v 1 , … , v k v_1, \ldots, v_k v 1 , … , v k , all starting at the origin at time t = 0 t = 0 t = 0 . Conjecture: for every runner i i i there exists a time t > 0 t > 0 t > 0 at which the (arc-length, circular) distance from runner i i i to every other runner is at least 1 / k 1/k 1/ k . Equivalently: for any k − 1 k - 1 k − 1 positive reals w 1 , … , w k − 1 w_1, \ldots, w_{k-1} w 1 , … , w k − 1 (the relative speeds), there exists t ≥ 0 t \geq 0 t ≥ 0 with ∥ t w j ∥ ≥ 1 / k \| t w_j \| \geq 1/k ∥ t w j ∥ ≥ 1/ k for all j j j , where ∥ ⋅ ∥ \|\cdot\| ∥ ⋅ ∥ is the distance to the nearest integer. 11
Erdős-Ko-Rado for cross-intersecting families Open
Fix integers n ≥ 2 k n \geq 2k n ≥ 2 k and t ≥ 1 t \geq 1 t ≥ 1 . Two families F , G ⊆ ( [ n ] k ) \mathcal{F}, \mathcal{G} \subseteq \binom{[n]}{k} F , G ⊆ ( k [ n ] ) are \emph{cross- t t t -intersecting} if ∣ F ∩ G ∣ ≥ t |F \cap G| \geq t ∣ F ∩ G ∣ ≥ t for every F ∈ F F \in \mathcal{F} F ∈ F and every G ∈ G G \in \mathcal{G} G ∈ G . \textbf{Conjecture (Frankl-Tokushige):} For all n ≥ ( t + 1 ) ( k − t + 1 ) n \geq (t+1)(k-t+1) n ≥ ( t + 1 ) ( k − t + 1 ) , ∣ F ∣ ⋅ ∣ G ∣ ≤ ( n − t k − t ) 2 |\mathcal{F}| \cdot |\mathcal{G}| \leq \binom{n-t}{k-t}^2 ∣ F ∣ ⋅ ∣ G ∣ ≤ ( k − t n − t ) 2 , with equality iff F = G = { F ∈ ( [ n ] k ) : T ⊆ F } \mathcal{F} = \mathcal{G} = \{F \in \binom{[n]}{k} : T \subseteq F\} F = G = { F ∈ ( k [ n ] ) : T ⊆ F } for some fixed t t t -set T T T . \textbf{Task:} Establish the conjecture in its full range of parameters ( n , k , t ) (n, k, t) ( n , k , t ) , including the regime currently open for small t t t . erdos-ko-rado-cross-intersecting
attack →