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Big Line or Big Clique in Planar Point Sets ★★

Author(s): Kara; Por; Wood

Let $ S $ be a set of points in the plane. Two points $ v $ and $ w $ in $ S $ are visible with respect to $ S $ if the line segment between $ v $ and $ w $ contains no other point in $ S $.

Conjecture   For all integers $ k,\ell\geq2 $ there is an integer $ n $ such that every set of at least $ n $ points in the plane contains at least $ \ell $ collinear points or $ k $ pairwise visible points.

Keywords: Discrete Geometry; Geometric Ramsey Theory

Hoàng-Reed Conjecture ★★★

Author(s): Hoang; Reed

Conjecture   Every digraph in which each vertex has outdegree at least $ k $ contains $ k $ directed cycles $ C_1, \ldots, C_k $ such that $ C_j $ meets $ \cup_{i=1}^{j-1}C_i $ in at most one vertex, $ 2 \leq j \leq k $.

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Hungry Shark World Cheats Generator IOS Android No Verification 2024 (fresh method) ★★

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Hungry Shark World Cheats Generator IOS Android No Verification 2024 (fresh method)

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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

5-flow conjecture ★★★★

Author(s): Tutte

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

Keywords: cubic; nowhere-zero flow

KPZ Universality Conjecture ★★★

Author(s):

Conjecture   Formulate a central limit theorem for the KPZ universality class.

Keywords: KPZ equation, central limit theorem

Odd incongruent covering systems ★★★

Author(s): Erdos; Selfridge

Conjecture   There is no covering system whose moduli are odd, distinct, and greater than 1.

Keywords: covering system

Cheats Free* Sims FreePlay Simoleons Life Points and Social Points Cheats 2024 No Human Verification ★★

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Cheats Free* Sims FreePlay Simoleons Life Points and Social Points Cheats 2024 No Human Verification

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Call Of Duty Mobile Generator Cheats No Human Verification (Without Surveys) ★★

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Call Of Duty Mobile Generator Cheats No Human Verification (Without Surveys)

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Jacob Palis Conjecture(Finitude of Attractors)(Dynamical Systems) ★★★★

Author(s):

Conjecture   Let $ Diff^{r}(M)  $ be the space of $ C^{r} $ Diffeomorphisms on the connected , compact and boundaryles manifold M and $ \chi^{r}(M) $ the space of $ C^{r} $ vector fields. There is a dense set $ D\subset Diff^{r}(M) $ ($ D\subset \chi^{r}(M) $ ) such that $ \forall f\in D $ exhibit a finite number of attractor whose basins cover Lebesgue almost all ambient space $ M $

This is a very Deep and Hard problem in Dynamical Systems . It present the dream of the dynamicist mathematicians .

Keywords: Attractors , basins, Finite

2-accessibility of primes ★★

Author(s): Landman; Robertson

Question   Is the set of prime numbers 2-accessible?

Keywords: monochromatic diffsequences; primes

The 4x5 chessboard complex is the complement of a link, which link? ★★

Author(s): David Eppstein

Problem   Ian Agol and Matthias Goerner observed that the 4x5 chessboard complex is the complement of many distinct links in the 3-sphere. Their observation is non-constructive, as it uses the resolution of the Poincare Conjecture. Find specific links that have the 4x5 chessboard complex as their complement.

Keywords: knot theory, links, chessboard complex

Universal point sets for planar graphs ★★★

Author(s): Mohar

We say that a set $ P \subseteq {\mathbb R}^2 $ is $ n $-universal if every $ n $ vertex planar graph can be drawn in the plane so that each vertex maps to a distinct point in $ P $, and all edges are (non-intersecting) straight line segments.

Question   Does there exist an $ n $-universal set of size $ O(n) $?

Keywords: geometric graph; planar graph; universal set

Cube-Simplex conjecture ★★★

Author(s): Kalai

Conjecture   For every positive integer $ k $, there exists an integer $ d $ so that every polytope of dimension $ \ge d $ has a $ k $-dimensional face which is either a simplex or is combinatorially isomorphic to a $ k $-dimensional cube.

Keywords: cube; facet; polytope; simplex

Legal Bleach Brave Souls Cheats Generator No Human Verification 2024 (No Surveys Needed) ★★

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Legal Bleach Brave Souls Cheats Generator No Human Verification 2024 (No Surveys Needed)

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Sums of independent random variables with unbounded variance ★★

Author(s): Feige

Conjecture   If $ X_1, \dotsc, X_n \geq 0 $ are independent random variables with $ \mathbb{E}[X_i] \leq \mu $, then $$\mathrm{Pr} \left( \sum X_i - \mathbb{E} \left[ \sum X_i \right ] < \delta \mu \right) \geq \min \left ( (1 + \delta)^{-1} \delta, e^{-1} \right).$$

Keywords: Inequality; Probability Theory; randomness in TCS

Geometry Dash Free Gold Coins Stars Cheats 2024 (FREE!) ★★

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Geometry Dash Free Gold Coins Stars Cheats 2024 (FREE!)

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Ádám's Conjecture ★★★

Author(s): Ádám

Conjecture   Every digraph with at least one directed cycle has an arc whose reversal reduces the number of directed cycles.

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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

Hamilton decomposition of prisms over 3-connected cubic planar graphs ★★

Author(s): Alspach; Rosenfeld

Conjecture   Every prism over a $ 3 $-connected cubic planar graph can be decomposed into two Hamilton cycles.

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Free Generator Matchington Mansion Working Stars Coins Cheats (Matchington Mansion Generator) ★★

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Free Generator Matchington Mansion Working Stars Coins Cheats (Matchington Mansion Generator)

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Is Skewes' number e^e^e^79 an integer? ★★

Author(s):

Conjecture  

Skewes' number $ e^{e^{e^{79}}} $ is not an integer.

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Polignac's Conjecture ★★★

Author(s): de Polignac

Conjecture   Polignac's Conjecture: For any positive even number n, there are infinitely many prime gaps of size n. In other words: There are infinitely many cases of two consecutive prime numbers with difference n.

In particular, this implies:

Conjecture   Twin Prime Conjecture: There are an infinite number of twin primes.

Keywords: prime; prime gap

Arc-disjoint directed cycles in regular directed graphs ★★

Author(s): Alon; McDiarmid; Molloy

Conjecture   If $ G $ is a $ k $-regular directed graph with no parallel arcs, then $ G $ contains a collection of $ {k+1 \choose 2} $ arc-disjoint directed cycles.

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Coloring the Odd Distance Graph ★★★

Author(s): Rosenfeld

The Odd Distance Graph, denoted $ {\mathcal O} $, is the graph with vertex set $ {\mathbb R}^2 $ and two points adjacent if the distance between them is an odd integer.

Question   Is $ \chi({\mathcal O}) = \infty $?

Keywords: coloring; geometric graph; odd distance

Crossing sequences ★★

Author(s): Archdeacon; Bonnington; Siran

Conjecture   Let $ (a_0,a_1,a_2,\ldots,0) $ be a sequence of nonnegative integers which strictly decreases until $ 0 $.

Then there exists a graph that be drawn on a surface with orientable (nonorientable, resp.) genus $ i $ with $ a_i $ crossings, but not with less crossings.

Keywords: crossing number; crossing sequence

Asymptotic Distribution of Form of Polyhedra ★★

Author(s): Rüdinger

Problem   Consider the set of all topologically inequivalent polyhedra with $ k $ edges. Define a form parameter for a polyhedron as $ \beta:= v/(k+2) $ where $ v $ is the number of vertices. What is the distribution of $ \beta $ for $ k \to \infty $?

Keywords: polyhedral graphs, distribution

Random stable roommates ★★

Author(s): Mertens

Conjecture   The probability that a random instance of the stable roommates problem on $ n \in 2{\mathbb N} $ people admits a solution is $ \Theta( n ^{-1/4} ) $.

Keywords: stable marriage; stable roommates

Legal* Free Coin Master Cheats Spins Coins Generator No Human Verification 2024 ★★

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Legal* Free Coin Master Cheats Spins Coins Generator No Human Verification 2024

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Direct proof of a theorem about compact funcoids ★★

Author(s): Porton

Conjecture   Let $ f $ is a $ T_1 $-separable (the same as $ T_2 $ for symmetric transitive) compact funcoid and $ g $ is a uniform space (reflexive, symmetric, and transitive endoreloid) such that $ ( \mathsf{\tmop{FCD}}) g = f $. Then $ g = \langle f \times f \rangle^{\ast} \Delta $.

The main purpose here is to find a direct proof of this conjecture. It seems that this conjecture can be derived from the well known theorem about existence of exactly one uniformity on a compact set. But that would be what I call an indirect proof, we need a direct proof instead.

The direct proof may be constructed by correcting all errors an omissions in this draft article.

Direct proof could be better because with it we would get a little more general statement like this:

Conjecture   Let $ f $ be a $ T_1 $-separable compact reflexive symmetric funcoid and $ g $ be a reloid such that
    \item $ ( \mathsf{\tmop{FCD}}) g = f $; \item $ g \circ g^{- 1} \sqsubseteq g $.

Then $ g = \langle f \times f \rangle^{\ast} \Delta $.

Keywords: compact space; compact topology; funcoid; reloid; uniform space; uniformity

Book Thickness of Subdivisions ★★

Author(s): Blankenship; Oporowski

Let $ G $ be a finite undirected simple graph.

A $ k $-page book embedding of $ G $ consists of a linear order $ \preceq $ of $ V(G) $ and a (non-proper) $ k $-colouring of $ E(G) $ such that edges with the same colour do not cross with respect to $ \preceq $. That is, if $ v\prec x\prec w\prec y $ for some edges $ vw,xy\in E(G) $, then $ vw $ and $ xy $ receive distinct colours.

One can think that the vertices are placed along the spine of a book, and the edges are drawn without crossings on the pages of the book.

The book thickness of $ G $, denoted by bt$ (G) $ is the minimum integer $ k $ for which there is a $ k $-page book embedding of $ G $.

Let $ G' $ be the graph obtained by subdividing each edge of $ G $ exactly once.

Conjecture   There is a function $ f $ such that for every graph $ G $, $$   \text{bt}(G) \leq f( \text{bt}(G') )\enspace.   $$

Keywords: book embedding; book thickness

Antidirected trees in digraphs ★★

Author(s): Addario-Berry; Havet; Linhares Sales; Reed; Thomassé

An antidirected tree is an orientation of a tree in which every vertex has either indegree 0 or outdergree 0.

Conjecture   Let $ D $ be a digraph. If $ |A(D)| > (k-2) |V(D)| $, then $ D $ contains every antidirected tree of order $ k $.

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Weighted colouring of hexagonal graphs. ★★

Author(s): McDiarmid; Reed

Conjecture   There is an absolute constant $ c $ such that for every hexagonal graph $ G $ and vertex weighting $ p:V(G)\rightarrow \mathbb{N} $, $$\chi(G,p) \leq \frac{9}{8}\omega(G,p) + c $$

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Outer reloid of restricted funcoid ★★

Author(s): Porton

Question   $ ( \mathsf{RLD})_{\mathrm{out}} (f \cap^{\mathsf{FCD}} ( \mathcal{A} \times^{\mathsf{FCD}} \mathcal{B})) = (( \mathsf{RLD})_{\mathrm{out}} f) \cap^{\mathsf{RLD}} ( \mathcal{A} \times^{\mathsf{RLD}} \mathcal{B}) $ for every filter objects $ \mathcal{A} $ and $ \mathcal{B} $ and a funcoid $ f\in\mathsf{FCD}(\mathrm{Src}\,f; \mathrm{Dst}\,f) $?

Keywords: direct product of filters; outer reloid

Kneser–Poulsen conjecture ★★★

Author(s): Kneser; Poulsen

Conjecture   If a finite set of unit balls in $ \mathbb{R}^n $ is rearranged so that the distance between each pair of centers does not decrease, then the volume of the union of the balls does not decrease.

Keywords: pushing disks

The Sims Mobile Cheats Generator Working Android Ios 2024 Cheats Generator (Newly Discovered) ★★

Author(s):

The Sims Mobile Cheats Generator Working Android Ios 2024 Cheats Generator (Newly Discovered)

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Lords Mobile Working Cheats Gems Coins Generator (NEW AND FREE) ★★

Author(s):

Lords Mobile Working Cheats Gems Coins Generator (NEW AND FREE)

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Colouring the square of a planar graph ★★

Author(s): Wegner

Conjecture   Let $ G $ be a planar graph of maximum degree $ \Delta $. The chromatic number of its square is
    \item at most $ 7 $ if $ \Delta =3 $, \item at most $ \Delta+5 $ if $ 4\leq\Delta\leq 7 $, \item at most $ \left\lfloor\frac32\,\Delta\right\rfloor+1 $ if $ \Delta\ge8 $.

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Solution to the Lonely Runner Conjecture ★★

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Solution to the Lonely Runner Conjecture

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List Hadwiger Conjecture ★★

Author(s): Kawarabayashi; Mohar

Conjecture   Every $ K_t $-minor-free graph is $ c t $-list-colourable for some constant $ c\geq1 $.

Keywords: Hadwiger conjecture; list colouring; minors

Family Island Cheats Generator 2024 No Human Verification (Real) ★★

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Family Island Cheats Generator 2024 No Human Verification (Real)

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Hedetniemi's Conjecture ★★★

Author(s): Hedetniemi

Conjecture   If $ G,H $ are simple finite graphs, then $ \chi(G \times H) = \min \{ \chi(G), \chi(H) \} $.

Here $ G \times H $ is the tensor product (also called the direct or categorical product) of $ G $ and $ H $.

Keywords: categorical product; coloring; homomorphism; tensor product

Free Coin Master Cheats No Human Verification No Survey (2024 Method) ★★

Author(s):

Free Coin Master Cheats No Human Verification No Survey (2024 Method)

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Extremal problem on the number of tree endomorphism ★★

Author(s): Zhicong Lin

Conjecture   An endomorphism of a graph is a mapping on the vertex set of the graph which preserves edges. Among all the $ n $ vertices' trees, the star with $ n $ vertices has the most endomorphisms, while the path with $ n $ vertices has the least endomorphisms.

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Partitionning a tournament into k-strongly connected subtournaments. ★★

Author(s): Thomassen

Problem   Let $ k_1, \dots , k_p $ be positve integer Does there exists an integer $ g(k_1, \dots , k_p) $ such that every $ g(k_1, \dots , k_p) $-strong tournament $ T $ admits a partition $ (V_1\dots , V_p) $ of its vertex set such that the subtournament induced by $ V_i $ is a non-trivial $ k_i $-strong for all $ 1\leq i\leq p $.

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Toon Blast Cheats Generator Android Ios 2024 Cheats Generator (re-designed) ★★

Author(s):

Toon Blast Cheats Generator Android Ios 2024 Cheats Generator (re-designed)

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Cooking Fever Cheats Generator Latest Version 2024 For Free (WORKING Generator) ★★

Author(s):

Cooking Fever Cheats Generator Latest Version 2024 For Free (WORKING Generator)

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P vs. PSPACE ★★★

Author(s): Folklore

Problem   Is there a problem that can be computed by a Turing machine in polynomial space and unbounded time but not in polynomial time? More formally, does P = PSPACE?

Keywords: P; PSPACE; separation; unconditional

Hungry Shark Evolution Cheats Generator 2024 Working (Generator) ★★

Author(s):

Hungry Shark Evolution Cheats Generator 2024 Working (Generator)

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trace inequality ★★

Author(s):

Let $ A,B $ be positive semidefinite, by Jensen's inequality, it is easy to see $ [tr(A^s+B^s)]^{\frac{1}{s}}\leq [tr(A^r+B^r)]^{\frac{1}{r}} $, whenever $ s>r>0 $.

What about the $ tr(A^s+B^s)^{\frac{1}{s}}\leq tr(A^r+B^r)^{\frac{1}{r}} $, is it still valid?

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