Catalog
Every problem here was open when it entered the pool. Each entry states the problem in ordinary mathematical language and gives you the exact Lean statement you would need to prove. Nothing is paraphrased, so what you read is what gets checked.
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Catalog
Every problem here was open when it entered the pool. Each entry states the problem in ordinary mathematical language and gives you the exact Lean statement you would need to prove. Nothing is paraphrased, so what you read is what gets checked.
Fields All fields (290) Logic and foundations (1) Combinatorics (61) Number theory (202) Group theory (3) Measure and integration (1) Complex functions (1) Dynamical systems (1) Sequences and series (1) Harmonic analysis (1) Geometry (2) Convex and discrete geometry (12) Probability (1) Information and communication (3) ▾
such that every integer is the sum of a prime and at most
powers of
?
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Bounty $2,927 Let A = { 1 , 2 , 4 , 8 , 13 , 21 , 31 , 45 , 66 , 81 , 97 , … } A = \{1, 2, 4, 8, 13, 21, 31, 45, 66, 81, 97, \ldots\} A = { 1 , 2 , 4 , 8 , 13 , 21 , 31 , 45 , 66 , 81 , 97 , … } be the greedy Sidon sequence: we begin with in the above formulation.
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Bounty $2,927
Bounty
$3,079
for some
a , b ∈ R a,b \in \mathbb{R} a , b ∈ R and
?
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Bounty $3,079 . Is it true that
1 t ∑ 1 ≤ i < t ( s i + 1 − s i ) 2 → ∞ \frac{1}{t}\sum_{1\leq i<t}(s_{i+1}-s_i)^2 \to \infty t 1 ∑ 1 ≤ i < t ( s i + 1 − s i ) 2 → ∞
as
∣ A ∣ → ∞ \lvert A\rvert\to \infty ∣ A ∣ → ∞ ?
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Bounty $3,079
for all sufficiently large
?
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Bounty $3,079
$3,079 and
. Show that
f ( n ) = o ( log n ) f(n)=o(\log n) f ( n ) = o ( log n ) .
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Bounty $3,079
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Bounty $3,079 such that
and repeat with
replaced by
. If this terminates after finitely many steps then this produces a representation of
as the sum…
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Bounty $3,079 and iteratively include the next smallest integer that preserves the Sidon property (i.e. there are no non-trivial solutions to
a + b = c + d a + b = c + d a + b = c + d ). What is the order of growth of
? Is it true that…
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Bounty $3,079 such that no subset of size
has the same pairwise greatest common divisor between all elements. Erdős [Er64] proved that
f 3 ( N ) > N c / log log N f_3(N) > N^{c/\log\log N} f 3 ( N ) > N c / l o g l o g N for some constant
, and conjectured this should also be an upper…
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Bounty $3,079 for all sufficiently large
.
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Bounty $3,079 -coloured then there is a monochromatic copy of the complete
-uniform
hypergraph on
vertices.
Is there some constant
such that
R 3 ( n ) ≥ 2 2 c n ? R_3(n) \geq 2^{2^{cn}}? R 3 ( n ) ≥ 2 2 c n ?
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Bounty $3,079 (the octahedron) and at least
edges, must
contain an independent set of size
? This is a problem of Erdős, Hajnal, Sós, and Szemerédi [EHSS83]. It is **open**; they proved the statement…
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Bounty $3,079 as
?
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Bounty $3,079 -coloured then there exist
vertices with at
least one colour missing on the edges of the induced
.
In other words, there is no balanced colouring.
A conjecture of Erdős and Gyárfás [ErGy99].
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Bounty $3,079 so that for every
with
∣ Y ∣ ≥ H ( n ) \lvert Y\rvert \geq H(n) ∣ Y ∣ ≥ H ( n )
we have
{ f ( A ) : A ⊆ Y } = X \left\{ f(A) : A\subseteq Y\right\}=X { f ( A ) : A ⊆ Y } = X .
Prove that
H ( n ) − log 2 n → ∞ H(n)-\log_2 n \to \infty H ( n ) − log 2 n → ∞ .
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Bounty $3,079 and let
R ( x i ) = # { ∣ x j − x i ∣ : j ≠ i } R(x_i)=\#\{ \lvert x_j-x_i\rvert : j\neq i\} R ( x i ) = # {∣ x j − x i ∣ : j = i } ,
where the points are ordered such that
R ( x 1 ) ≤ ⋯ ≤ R ( x n ) . R(x_1)\leq \cdots \leq R(x_n). R ( x 1 ) ≤ ⋯ ≤ R ( x n ) .
Let
be the maximum number of distinct values the
can take. Is it true that
g ( n ) ≥ ( 1 − o ( 1 ) ) n g(n) \geq (1-o(1))n g ( n ) ≥ ( 1 − o ( 1 )) n ?
such that whenever
F ′ ⊆ F \mathcal{F}'\subseteq \mathcal{F} F ′ ⊆ F is an intersecting subfamily we have…
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Bounty $3,079 with
∣ B ∣ ≥ h ( n ) \lvert B\rvert \geq h(n) ∣ B ∣ ≥ h ( n ) such that if
a 1 + ⋯ + a r = b 1 + ⋯ + b s a_1+\cdots+a_r=b_1+\cdots+b_s a 1 + ⋯ + a r = b 1 + ⋯ + b s with
a i , b i ∈ B a_i,b_i\in B a i , b i ∈ B then
.
Is
h ( n ) = Θ ( n ) h(n) = \Theta(\sqrt{n}) h ( n ) = Θ ( n ) ?
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Bounty $3,079 , for all large
?
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Bounty $3,079 with
∣ A ∣ = k + 1 \lvert A\rvert=k+1 ∣ A ∣ = k + 1 all
colours appear among the
-sized subsets of
?
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Bounty $3,079 tend to infinity?
(Other finite limits have been ruled out by [KoLu25], see below)
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Bounty $3,079 of cardinality continuum such that
A + A ⊆ R ∖ S A + A \subseteq \mathbb{R}\setminus S A + A ⊆ R ∖ S ?
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Bounty $2,927 (see…
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Bounty $2,927 for all
large
) such that
∑ n ≤ x f r ( n ) 2 ≪ x \sum_{n\leq x}f_r(n)^2 \ll x ∑ n ≤ x f r ( n ) 2 ≪ x for all
?
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Bounty $3,079 , where
?
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Bounty $3,079 tuples
( x 1 , … , x 5 , y 1 , … , y 5 ) ∈ G 10 (x_1, \dots, x_5, y_1, \dots, y_5) \in G^{10} ( x 1 , … , x 5 , y 1 , … , y 5 ) ∈ G 10
such that
x i + y j ∈ A x_i + y_j \in A x i + y j ∈ A whenever
j ∈ { i , i + 1 , i + 2 } j \in \{i, i+1, i+2\} j ∈ { i , i + 1 , i + 2 } ?
Note: We interpret indices modulo 5.
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Bounty $3,079 is free of 3-term progressions?
triples
such that
( x , y ) , ( g x , y ) , ( x , g y ) (x, y), (gx, y), (x, gy) ( x , y ) , ( g x , y ) , ( x , g y )
all lie in
?
Note: A is taken as
-dense, i.e.
∣ A ∣ ≥ α ∣ G ∣ 2 |A| \ge \alpha |G|^2 ∣ A ∣ ≥ α ∣ G ∣ 2 [Au16, Question 2]
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Bounty $3,079 .
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Bounty $3,079 .
Is there a dilate of
containing a gap of length
?
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Bounty $3,079 , with
A + A = Z / q Z A + A = \mathbb{Z}/q\mathbb{Z} A + A = Z / q Z ? [Gr24]
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Bounty $3,079 , can we almost surely cover
Z / p Z \mathbb{Z}/p\mathbb{Z} Z / p Z with
translates of
? [Gr24]
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Bounty $2,927 contain a coset of some subspace of dimension at least
n − O ( log ( 1 / α ) ) n - O(\log(1/\alpha)) n − O ( log ( 1/ α )) ? More precisely: does there exist an absolute constant
such that for all
and all nonempty
A ⊆ F 2 n A \subseteq \mathbb{F}_2^n A ⊆ F 2 n with density
…
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Bounty $3,079 .
Does
contain a subspace of co-dimension
? [Sa11, Question 5.1]
contain a composite number?
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Bounty $3,079
Open problems · Conjectures.io