... Annals of Mathematics, 157 (2003), 545–556 ** On ** ** a ** ** coloring ** ** conjecture ** ** about ** ** unit ** ** fractions ** By Ernest S Croot III Abstract We prove an old ** conjecture ** of Erd˝s and Graham ** on ** sums of ** unit ** ** fractions:** ... we have that (1.1) follows = o(r), n 549 ** ON ** ** A ** ** COLORING ** ** CONJECTURE ** ** ABOUT ** ** UNIT ** ** FRACTIONS ** Technical lemmas and their proofs Lemma If w1 and w2 are distinct integers which both lie in an interval ... Erd˝s and Graham, which appears in [2], [3], and [5] o We will need to introduce some notation and deﬁnitions in order to state the Main Theorem, as well as the propositions and lemmas in later...

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... & Co Inc ** and ** the Otho S.A Sprague Memorial Institute Breathing can be okay Your ** asthma ** can be well controlled This ** coloring ** ** and ** ** activity ** ** book ** is for children ** and ** their families Each ** activity ** ... people who are dying from ** asthma ** is going up • ** Asthma ** is expensive for the United States Missed work ** and ** school due to ** asthma,** ** asthma ** medicines ** and ** hospital visits for ** asthma ** cost $6,000,000,000 ... to understanding how to be your best with ** asthma ** It will tell about ** asthma ** ** and ** the plan created by you ** and ** your doctor There are pages to color, pictures to draw, things to figure out ** and ** puzzles...

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... ordinary online ** coloring ** of an unknown graph They constructed a class of log n-colorable graphs that require at least n/ log n colors online in the worst case [6] The best performance ratios ** known ** are ... Gy´rf´s and Lehel [4] a a holds also for our model of a ** known ** input graph Bartal et al [2] considered a diﬀerent version of online ** coloring ** a ** known ** graph In their model, each presented vertex is ... independence number α are ** known ** in advance The unknown case can be handled via a doubling technique, see [5] Without loss of generality, α ≥ n/ log n Let q = (1/2) logn/α n Before ** coloring,** randomly...

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... strictly increasing in s An easy calculation shows F (Z∞ ** **** ) ** = F (Z∞ ** **** ) ** = (1/16 **)n2** ** **** + ** O **(n)** And The Winner Is: Z0 = 0 **4n/** 11 1 **6n/** 11 **0n/** 11 setting the world-record ** of ** (1 **/22** **)n2** ** **** + ** O **(n)** Note: Tomasz Schoen[S], ... < ** n/** (12s ** **** + ** ** **** ****2)** Case I: If ** n/** (12s ** **** + ** ** ****8) ** ≤ w < ** n/** (12s ** **** + ** ** **** ****2)** then the unique solution is ** n ** n− ** n ** −w−1 w+1 0 s−1 ** n ** n 4w 6w−1 6w−1 s−1 (1 ** **** ) ** ** 2 ** −(6s ** **** ****2)** w+s−1 **0n** −(6s+1)w+s−1 16w−1 (06w−1 16w−1 ** **** ) ** 0w+1 ... 1 **/2 ** ** **** + ** and if ** n ** is odd then z **(n+** ** ****1) ** **/2 ** = H1 **/2 ** (k − ** n ** ** **** + ** ** n** r ** n ** (Right V olley) ** n** r+1 zj , − j=r ** n+** 1 ** **** + ** (Lef t V olley) **(n** ** ****1) ** **/2 ** zj ** **** ) ** j=1 These equations uniquely determine z (if it exists), in...

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... + t2 ≥ 2t1 Suppose t1 + t2 < **2n/** 3 Then **2n/** 3 > ** n ** − t1 (so t1 > ** n/** 3) and 2t1 < **2n/** 3 which yields the contradiction **2n/** 3 > ** n ** − ** n/** 3 Hence t1 + t2 ≥ **2n/** 3 Finally we see that the size of f is ** n(** n ... the ** **** × ** ** n ** array is f -choosable So the size of f is at least ** n(** n + 1) **/2 ** and hence ** n ** i=1 f **(2,** i) ≥ **(n ** − 2t1 ) + ** n(** n + 1) **/2 ** So t2 ≥ ** n ** − 2t1 and t1 ≤ t2 Thus t1 + t2 ≥ t1 + **(n ** − 2t1 ) = ** n ** − t1 and ... is ** n(** n + 1) **/2 ** + t1 + ** n(** n + 1) **/2 ** + t2 ≥ ** n(** n + 1) + **2n/** 3 = ** n2** + **5n/** 3 the electronic journal of combinatorics **(20** **02)** , **#N8** Upper Bound To complete the proof of Theorem we need to construct a choosable...

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... Cytology ** and ** Genetics of Russian Academy of Sciences for support ** and ** hospitality, ** and ** to D Fon Der Flaass for useful comments References [1] Albertson, M., Open questions in ** Graph ** Color ** Extensions,** ... The listcoloring version of Brooks’ theorem was considered much earlier by Vizing [5] We need a couple of deﬁnitions ﬁrst A block containing an edge e is a maximum 2-connected subgraph containing ... tree ** and ** |l(x)| = d(x) for each x ∈ V Figure depicts graphs illustrating the exactness of our results Next we give a formal description of ** graph ** G1 from the ﬁgure A general construction Consider...

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... w ** with ** the same color Now the vertex v between u, w on P4 is not in C, but since u, w are at distance at least on C the path uvw together ** with ** one of the u, w-segments of C yields a shorter non-contractible ... since otherwise it would contain a ** 2-colored ** P4 ) We will use the following terminology Deﬁnition 2.1 An r-coloring of G is called a star coloring if there are ** no ** ** 2-colored ** paths on vertices The ... this ** with ** an example in which vertices in X are denoted by ⊗ and those in Y by • Edges not in the cut are denoted by dotted lines and edges in F ** with ** double lines the electronic journal of combinatorics...

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... quadrilateralization and ** a ** 4 **-coloring ** ** of ** the resultant quadrilateralization graph Let us recall that ** a ** quadrilateralization ** of ** an orthogonal polygon P **(with ** or without holes) is ** a ** partitioning ** of ** ... quadrilateralization, and the dual graph GD (b) Quadrilateralization graph GQ **(a)** (b) skewed quadrilateral balanced quadrilateral Figure 4: **(a)** Graph Gk (b) Balanced and skewed quadrilaterals ... the dual graph ** of ** quadrilateralization Q: each vertex ** of ** GD corresponds to ** a ** quadrilateral and two vertices are adjacent if their quadrilaterals share ** a ** side Clearly, the dual graph GD is ** a ** single...

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... deﬁne ** oriented ** arc-colorings ** of ** ** oriented ** graphs in a natural way by saying that, as in the undirected case, an ** oriented ** ** arc-coloring ** ** of ** an ** oriented ** graph G is an ** oriented ** vertex-coloring ** of ** the ... arcs ** of ** A Let G be an ** oriented ** graph and f be an ** oriented ** ** arc-coloring ** ** of ** G For a given vertex + − v ** of ** G, we denote by Cf (v) and Cf (v) the outgoing color set ** of ** v (i.e the set ** of ** colors ** of ** ... due to the minimality ** of ** H, there exists a good QR7 **-arc-coloring ** f ** of ** H The coloring f is a partial good QR7 **-arc-coloring ** ** of ** H, that is an ** arc-coloring ** ** of ** some subset S ** of ** A(H) and we show how...

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... that every ** planar ** graph of ** girth ** at least can be star colored using 16 colors, every ** planar ** graph of ** girth ** at least can be star colored with colors, and ** planar ** graphs of suﬃciently large ** girth ** can ... combinatorics 15 (2008), #R124 4 ** Girth ** ** planar ** graphs To prove that ** girth ** ** planar ** graphs can be star colored with colors, we use a similar approach as used for ** girth ** 14 ** planar ** graphs, except that the ... upper bounds for ** planar ** graphs of ** high ** ** girth,** less is known about ** planar ** graphs of low ** girth ** As mentioned in the introduction, Albertson et al [1] show the star chromatic number for ** planar ** graphs...

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... k → ∞ If ** the ** numerator of ** the ** other quantity does not tend to ∞, then we are done since ** the ** denominator does tend to ∞ Otherwise, we can use L’hopital’s rule, from which we get that ** the ** second ... increases Rk−1 from to π Also, ** the ** signs of these integrals alternate, so we can either throw out all r of them or all but ** the ** ﬁrst one, depending on whether ** the ** integral of h across R⌊ π ⌋ ... establishes that ** the ** measurable chromatic number of ** the ** ** odd-distance ** graph is inﬁnite Conclusion and Open Problems ** The ** largest remaining question is whether or not ** the ** chromatic number in ** the ** normal...

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... orientation ** of ** edges from E does not matter Figure 2: Coloring ** of ** the subdivision Take any simple path P from the ** subdivided ** graph G′ and let P be the word consisting the ** of ** the labels ** of ** the vertices ... 212 as a substring Proof For 010 to appear as a substring ** of ** some βi , 00100 must be a subword ** of ** γi As γi is constructed from γi−1 by substituting a single digit by a pair ** of ** diﬀerent digits, ... necessary preparation in Section Figure 1: Non-repetitive ** 3-coloring ** ** of ** a ** subdivided ** clique K4 For other interesting problems on Thue colorings ** of ** graphs, see [AGHR02] Preliminaries First, we introduce...

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... 1.1 List ** coloring ** graphs The notion of list **-coloring ** is a generalization of the notion of proper ** coloring,** and has been studied extensively for graphs ... is explicitly displayed 1.2 List ** coloring ** hypergraphs There seem to be very few results on list colorings of hypergraphs Perhaps the most famous question on colorings of hypergraphs is the Erd˝s-Faber-Lov´sz ... that for a ﬁxed ** coloring ** χ of X, c P (Bχ ) < exp − |Y | 8(8s)s Since there are at most s|X| colorings of X, and since by Claim |Y | pn/2 and |X| c 2p2 n, the expected number of colorings χ of...

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