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P vs. BPP ★★★

Author(s): Folklore

Conjecture   Can all problems that can be computed by a probabilistic Turing machine (with error probability < 1/3) in polynomial time be solved by a deterministic Turing machine in polynomial time? That is, does P = BPP?

Keywords: BPP; circuit complexity; pseudorandom generators

Subgroup formed by elements of order dividing n ★★

Author(s): Frobenius

Conjecture  

Suppose $ G $ is a finite group, and $ n $ is a positive integer dividing $ |G| $. Suppose that $ G $ has exactly $ n $ solutions to $ x^{n} = 1 $. Does it follow that these solutions form a subgroup of $ G $?

Keywords: order, dividing

The Double Cap Conjecture ★★

Author(s): Kalai

Conjecture   The largest measure of a Lebesgue measurable subset of the unit sphere of $ \mathbb{R}^n $ containing no pair of orthogonal vectors is attained by two open caps of geodesic radius $ \pi/4 $ around the north and south poles.

Keywords: combinatorial geometry; independent set; orthogonality; projective plane; sphere

Saturation in the Hypercube ★★

Author(s): Morrison; Noel; Scott

Question   What is the saturation number of cycles of length $ 2\ell $ in the $ d $-dimensional hypercube?

Keywords: cycles; hypercube; minimum saturation; saturation

Rank vs. Genus ★★★

Author(s): Johnson

Question   Is there a hyperbolic 3-manifold whose fundamental group rank is strictly less than its Heegaard genus? How much can the two differ by?

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War Thunder Golden Eagles Generator Working Cheats (refreshed version) ★★

Author(s):

War Thunder Golden Eagles Generator Working Cheats (refreshed version)

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Complete bipartite subgraphs of perfect graphs ★★

Author(s): Fox

Problem   Let $ G $ be a perfect graph on $ n $ vertices. Is it true that either $ G $ or $ \bar{G} $ contains a complete bipartite subgraph with bipartition $ (A,B) $ so that $ |A|, |B| \ge n^{1 - o(1)} $?

Keywords: perfect graph

Clash of Clans Gems Cheats without verification (Free) ★★

Author(s):

Clash of Clans Gems Cheats without verification (Free)

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Sidorenko's Conjecture ★★★

Author(s): Sidorenko

Conjecture   For any bipartite graph $ H $ and graph $ G $, the number of homomorphisms from $ H $ to $ G $ is at least $ \left(\frac{2|E(G)|}{|V(G)|^2}\right)^{|E(H)|}|V(G)|^{|V(H)|} $.

Keywords: density problems; extremal combinatorics; homomorphism

Smooth 4-dimensional Poincare conjecture ★★★★

Author(s): Poincare; Smale; Stallings

Conjecture   If a $ 4 $-manifold has the homotopy type of the $ 4 $-sphere $ S^4 $, is it diffeomorphic to $ S^4 $?

Keywords: 4-manifold; poincare; sphere

The permanent conjecture ★★

Author(s): Kahn

Conjecture   If $ A $ is an invertible $ n \times n $ matrix, then there is an $ n \times n $ submatrix $ B $ of $ [A A] $ so that $ perm(B) $ is nonzero.

Keywords: invertible; matrix; permanent

Unused Free Kim Kardashian Hollywood Cheats No Human Verification No Survey (2024 Method) ★★

Author(s):

Unused Free Kim Kardashian Hollywood Cheats No Human Verification No Survey (2024 Method)

Keywords:

Real Racing 3 Cheats Generator Tested on iOS and Android (Latest Method) ★★

Author(s):

Real Racing 3 Cheats Generator Tested on iOS and Android (Latest Method)

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Multicolour Erdős--Hajnal Conjecture ★★★

Author(s): Erdos; Hajnal

Conjecture   For every fixed $ k\geq2 $ and fixed colouring $ \chi $ of $ E(K_k) $ with $ m $ colours, there exists $ \varepsilon>0 $ such that every colouring of the edges of $ K_n $ contains either $ k $ vertices whose edges are coloured according to $ \chi $ or $ n^\varepsilon $ vertices whose edges are coloured with at most $ m-1 $ colours.

Keywords: ramsey theory

Every 4-connected toroidal graph has a Hamilton cycle ★★

Author(s): Grunbaum; Nash-Williams

Conjecture   Every 4-connected toroidal graph has a Hamilton cycle.

Keywords:

Number of Cliques in Minor-Closed Classes ★★

Author(s): Wood

Question   Is there a constant $ c $ such that every $ n $-vertex $ K_t $-minor-free graph has at most $ c^tn $ cliques?

Keywords: clique; graph; minor

Golf Battle Free Cheats Generator 999,999k Free 2024 (Free Generator) ★★

Author(s):

Golf Battle Free Cheats Generator 999,999k Free 2024 (Free Generator)

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Length of surreal product

Author(s): Gonshor

Conjecture   Every surreal number has a unique sign expansion, i.e. function $ f: o\rightarrow \{-, +\} $, where $ o $ is some ordinal. This $ o $ is the length of given sign expansion and also the birthday of the corresponding surreal number. Let us denote this length of $ s $ as $ \ell(s) $.

It is easy to prove that

$$ \ell(s+t) \leq \ell(s)+\ell(t) $$

What about

$$ \ell(s\times t) \leq \ell(s)\times\ell(t) $$

?

Keywords: surreal numbers

The stubborn list partition problem ★★

Author(s): Cameron; Eschen; Hoang; Sritharan

Problem   Does there exist a polynomial time algorithm which takes as input a graph $ G $ and for every vertex $ v \in V(G) $ a subset $ \ell(v) $ of $ \{1,2,3,4\} $, and decides if there exists a partition of $ V(G) $ into $ \{A_1,A_2,A_3,A_4\} $ so that $ v \in A_i $ only if $ i \in \ell(v) $ and so that $ A_1,A_2 $ are independent, $ A_4 $ is a clique, and there are no edges between $ A_1 $ and $ A_3 $?

Keywords: list partition; polynomial algorithm

Real Racing 3 Cheats Generator Working 2024 (Real Racing 3 Generator) ★★

Author(s):

Real Racing 3 Cheats Generator Working 2024 (Real Racing 3 Generator)

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Cookie Run Kingdom Cheats Generator Android Ios 2024 Cheats Generator (free) ★★

Author(s):

Cookie Run Kingdom Cheats Generator Android Ios 2024 Cheats Generator (free)

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A homomorphism problem for flows ★★

Author(s): DeVos

Conjecture   Let $ M,M' $ be abelian groups and let $ B \subseteq M $ and $ B' \subseteq M' $ satisfy $ B=-B $ and $ B' = -B' $. If there is a homomorphism from $ Cayley(M,B) $ to $ Cayley(M',B') $, then every graph with a B-flow has a B'-flow.

Keywords: homomorphism; nowhere-zero flow; tension

The Alon-Tarsi basis conjecture ★★

Author(s): Alon; Linial; Meshulam

Conjecture   If $ B_1,B_2,\ldots B_p $ are invertible $ n \times n $ matrices with entries in $ {\mathbb Z}_p $ for a prime $ p $, then there is a $ n \times (p-1)n $ submatrix $ A $ of $ [B_1 B_2 \ldots B_p] $ so that $ A $ is an AT-base.

Keywords: additive basis; matrix

Growth of finitely presented groups ★★★

Author(s): Adyan

Problem   Does there exist a finitely presented group of intermediate growth?

Keywords: finitely presented; growth

Several ways to apply a (multivalued) multiargument function to a family of filters ★★★

Author(s): Porton

Problem   Let $ \mathcal{X} $ be an indexed family of filters on sets. Which of the below items are always pairwise equal?

1. The funcoid corresponding to this function (considered as a single argument function on indexed families) applied to the reloidal product of filters $ \mathcal{X} $.

2. The funcoid corresponding to this function (considered as a single argument function on indexed families) applied to the starred reloidal product of filters $ \mathcal{X} $.

3. $ \bigcap_{F\in\operatorname{up}^{\mathrm{FCD}}\prod^{\mathrm{Strd}}\mathcal{X}}\langle f \rangle F $.

Keywords: funcoid; function; multifuncoid; staroid

Odd perfect numbers ★★★

Author(s): Ancient/folklore

Conjecture   There is no odd perfect number.

Keywords: perfect number

Inverse Galois Problem ★★★★

Author(s): Hilbert

Conjecture   Every finite group is the Galois group of some finite algebraic extension of $ \mathbb Q $.

Keywords:

The circular embedding conjecture ★★★

Author(s): Haggard

Conjecture   Every 2-connected graph may be embedded in a surface so that the boundary of each face is a cycle.

Keywords: cover; cycle

Turán number of a finite family. ★★

Author(s): Erdos; Simonovits

Given a finite family $ {\cal F} $ of graphs and an integer $ n $, the Turán number $ ex(n,{\cal F}) $ of $ {\cal F} $ is the largest integer $ m $ such that there exists a graph on $ n $ vertices with $ m $ edges which contains no member of $ {\cal F} $ as a subgraph.

Conjecture   For every finite family $ {\cal F} $ of graphs there exists an $ F\in {\cal F} $ such that $ ex(n, F ) = O(ex(n, {\cal F})) $ .

Keywords:

Counterexamples to the Baillie-PSW primality test ★★

Author(s):

Problem  (1)   Find a counterexample to Baillie-PSW primality test or prove that there is no one.
Problem  (2)   Find a composite $ n\equiv 3 $ or $ 7\pmod{10} $ which divides both $ 2^{n-1} - 1 $ (see Fermat pseudoprime) and the Fibonacci number $ F_{n+1} $ (see Lucas pseudoprime), or prove that there is no such $ n $.

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Critical Ops Credits Cheats 2024 New Working Generator (New Method!) ★★

Author(s):

Critical Ops Credits Cheats 2024 New Working Generator (New Method!)

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Matchington Mansion Free Stars Coins Cheats Free Generator 2024 in 5 minutes (successive cheats) ★★

Author(s):

Matchington Mansion Free Stars Coins Cheats Free Generator 2024 in 5 minutes (successive cheats)

Keywords:

REAL* Free!! Call Of Duty Mobile Cheats Generator (Trick 2024) ★★

Author(s):

REAL* Free!! Call Of Duty Mobile Cheats Generator (Trick 2024)

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Lords Mobile Gems Coins Cheats Mod Android Ios No Survey 2024 (NEW) ★★

Author(s):

Lords Mobile Gems Coins Cheats Mod Android Ios No Survey 2024 (NEW)

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Sum of prime and semiprime conjecture ★★

Author(s): Geoffrey Marnell

Conjecture   Every even number greater than $ 10 $ can be represented as the sum of an odd prime number and an odd semiprime .

Keywords: prime; semiprime

Switching reconstruction conjecture ★★

Author(s): Stanley

Conjecture   Every simple graph on five or more vertices is switching-reconstructible.

Keywords: reconstruction

Free Call Of Duty Mobile Cheats Generator No Human Verification No Survey (Unused) ★★

Author(s):

Free Call Of Duty Mobile Cheats Generator No Human Verification No Survey (Unused)

Keywords:

Simpsons Tapped Out Cheats Generator Unlimited Cheats Generator (New 2024) ★★

Author(s):

Simpsons Tapped Out Cheats Generator Unlimited Cheats Generator (New 2024)

Keywords:

War Dragons Rubies Cheats 2024 (rejuvenated cheats) ★★

Author(s):

War Dragons Rubies Cheats 2024 (rejuvenated cheats)

Keywords:

Birch & Swinnerton-Dyer conjecture ★★★★

Author(s):

Conjecture   Let $ E/K $ be an elliptic curve over a number field $ K $. Then the order of the zeros of its $ L $-function, $ L(E, s) $, at $ s = 1 $ is the Mordell-Weil rank of $ E(K) $.

Keywords:

5-flow conjecture ★★★★

Author(s): Tutte

Conjecture   Every bridgeless graph has a nowhere-zero 5-flow.

Keywords: cubic; nowhere-zero flow

Square achievement game on an n x n grid ★★

Author(s): Erickson

Problem   Two players alternately write O's (first player) and X's (second player) in the unoccupied cells of an $ n \times n $ grid. The first player (if any) to occupy four cells at the vertices of a square with horizontal and vertical sides is the winner. What is the outcome of the game given optimal play? Note: Roland Bacher and Shalom Eliahou proved that every 15 x 15 binary matrix contains four equal entries (all 0's or all 1's) at the vertices of a square with horizontal and vertical sides. So the game must result in a winner (the first player) when n=15.

Keywords: game

A funcoid related to directed topological spaces ★★

Author(s): Porton

Conjecture   Let $ R $ be the complete funcoid corresponding to the usual topology on extended real line $ [-\infty,+\infty] = \mathbb{R}\cup\{-\infty,+\infty\} $. Let $ \geq $ be the order on this set. Then $ R\sqcap^{\mathsf{FCD}}\mathord{\geq} $ is a complete funcoid.
Proposition   It is easy to prove that $ \langle R\sqcap^{\mathsf{FCD}}\mathord{\geq}\rangle \{x\} $ is the infinitely small right neighborhood filter of point $ x\in[-\infty,+\infty] $.

If proved true, the conjecture then can be generalized to a wider class of posets.

Keywords:

Forcing a $K_6$-minor ★★

Author(s): Barát ; Joret; Wood

Conjecture   Every graph with minimum degree at least 7 contains a $ K_6 $-minor.
Conjecture   Every 7-connected graph contains a $ K_6 $-minor.

Keywords: connectivity; graph minors

Which compact boundaryless 3-manifolds embed smoothly in the 4-sphere? ★★★

Author(s): Kirby

Problem   Determine a computable set of invariants that allow one to determine, given a compact boundaryless 3-manifold, whether or not it embeds smoothly in the 4-sphere. This should include a constructive procedure to find an embedding if the manifold is embeddable.

Keywords: 3-manifold; 4-sphere; embedding

Chromatic number of associahedron ★★

Author(s): Fabila-Monroy; Flores-Penaloza; Huemer; Hurtado; Urrutia; Wood

Conjecture   Associahedra have unbounded chromatic number.

Keywords: associahedron, graph colouring, chromatic number

The Two Color Conjecture ★★

Author(s): Neumann-Lara

Conjecture   If $ G $ is an orientation of a simple planar graph, then there is a partition of $ V(G) $ into $ \{X_1,X_2\} $ so that the graph induced by $ X_i $ is acyclic for $ i=1,2 $.

Keywords: acyclic; digraph; planar

One-way functions exist ★★★★

Author(s):

Conjecture   One-way functions exist.

Keywords: one way function

Brawlhalla Cheats Generator 2024 (safe and working) ★★

Author(s):

Brawlhalla Cheats Generator 2024 (safe and working)

Keywords:

War Machines Cheats Free Unlimited Coins Diamonds Generator (new codes cheat) ★★

Author(s):

Conjecture  

Keywords: