Category
Problem Set
Status
The Erdős-Faber-Lovász Conjecture
If $n$ complete graphs, each with $n$ vertices, have the property that every pair of complete graphs shares at most one vertex, can the entire graph b...
The Union-Closed Sets Conjecture
For any finite family of finite sets that is closed under taking unions, must there exist an element that belongs to at least half of the sets?...
Singmaster's Conjecture
Does there exist a finite upper bound on how many times a number (other than 1) can appear in Pascal's triangle?...
The Keller Conjecture
Can every tiling of $\mathbb{R}^n$ by unit hypercubes have two cubes that share a complete $(n-1)$-dimensional face?...
The Kahn-Kalai Conjecture
For a monotone graph property, is the threshold for a random graph to have this property at most a constant factor away from the expectation threshold...
The Alon-Saks-Seymour Conjecture
Is the chromatic number of a graph at most its clique cover number times the maximum chromatic number of its neighborhoods?...
The Cameron-Erdős Conjecture
Is the number of sum-free subsets of $\{1, 2, \ldots, n\}$ equal to $O(2^{n/2})$?...
The Cap Set Problem
What is the maximum size of a cap set in $\mathbb{F}_3^n$?...
The Sunflower Conjecture
Does every family of at least $c^k k!$ sets of size $k$ contain a sunflower of size 3, for some absolute constant $c$?...
Ramsey Number $R(5,5)$
What is the exact value of the Ramsey number $R(5,5)$?...
1/3–2/3 Conjecture
Does every non-totally-ordered finite poset have two elements with probability between 1/3 and 2/3 in random linear extensions?...
Lonely Runner Conjecture
If $k$ runners with distinct speeds run on a circular track, will each be lonely (distance $\geq 1/k$ from others) at some time?...
Union-Closed Sets Conjecture
For a finite family of sets closed under unions, must some element appear in at least half the sets?...
No-Three-in-Line Problem
What is the maximum number of points in an $n \times n$ grid with no three collinear?...
Sunflower Conjecture
For fixed $r$, can the number of size-$k$ sets needed for an $r$-sunflower be bounded by $c^k$ for some constant $c$?...
Sudoku: Unique Solution Puzzles
How many Sudoku puzzles have exactly one solution?...
Sudoku: Minimal Puzzles Count
How many Sudoku puzzles with exactly one solution are minimal (removing any clue creates multiple solutions)?...
Maximum Givens in Minimal Sudoku
What is the maximum number of givens for a minimal Sudoku puzzle?...
Tic-Tac-Toe Winning Dimension
Given the width of a tic-tac-toe board, what is the smallest dimension guaranteeing X has a winning strategy?...
Perfect Chess
What is the outcome of a perfectly played game of chess?...
Perfect Komi in Go
What is the perfect value of komi (compensation points) in Go?...
Cap Set Problem
What is the largest possible cap set in $n$-dimensional affine space over the three-element field?...
Octal Games Periodicity
Are the nim-sequences of all finite octal games eventually periodic?...
Grundy's Game Periodicity
Is the nim-sequence of Grundy's game eventually periodic?...
Rendezvous Problem
What is the optimal strategy for two agents to meet on a network without communication?...
1/3-2/3 Conjecture
Does every non-total finite poset have two elements x,y with P(x before y in random linear extension) ∈ [1/3, 2/3]?...
Lonely Runner Conjecture
If k runners with distinct speeds run on a unit circle, will each runner be "lonely" (≥1/k away from others) at some time?...
Sunflower Conjecture
Can the minimum size for sunflowers be bounded by an exponential (not super-exponential) function of k?...
Union-Closed Sets Conjecture
For any finite union-closed family of sets, does some element appear in at least half the sets?...
Ramsey Number R(5,5)
What is the exact value of the Ramsey number R(5,5)?...