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A Conjecture of Erdös the Ramsey Number r(W6)
 J. Combinatorial Math. and Combinatorial Computing
, 1996
"... It was conjectured by Paul Erdos that if G is a graph with chromatic number at least k; then the diagonal Ramsey number r(G) r(K k ). That is, the complete graph K k has the smallest diagonal Ramsey number among the graphs of chromatic number k. This conjecture is shown to be false for k = 4 by ve ..."
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It was conjectured by Paul Erdos that if G is a graph with chromatic number at least k; then the diagonal Ramsey number r(G) r(K k ). That is, the complete graph K k has the smallest diagonal Ramsey number among the graphs of chromatic number k. This conjecture is shown to be false for k = 4 by verifying that r(W 6 ) = 17; where W 6 is the wheel with 6 vertices, since it is well known that r(K 4 ) = 18. Computational techniques are used to determine r(W 6 ) as well as the Ramsey numbers for other pairs of small order wheels. 1 Introduction The following well known conjecture is due to Paul Erdos. CONJECTURE 1 If G is a graph with chromatic number Ø(G) k; then the Ramsey number r(G) r(K k ): The strong form of the Erdos conjecture is that if Ø(G) k; and G does not contain a copy of K k ; then r(G) ? r(K k ). For k = 3 it is trivial to verify this stronger conjecture. If G 6' K 3 and Ø(G) 3; then G has at least 4 vertices. Thus r(G) ? 6 = r(K 3 ); since neither the graph K 3 [K...
Computing the Folkman Number
"... For a graph G, the expression G v → (a1,..., ar) means that for any rcoloring of the vertices of G there exists a monochromatic aiclique in G for some color i ∈ {1,..., r}. The vertex Folkman numbers are defined as Fv(a1,..., ar; q) = min{V (G)  : G v → (a1,..., ar) and Kq * G}. Of these, the o ..."
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For a graph G, the expression G v → (a1,..., ar) means that for any rcoloring of the vertices of G there exists a monochromatic aiclique in G for some color i ∈ {1,..., r}. The vertex Folkman numbers are defined as Fv(a1,..., ar; q) = min{V (G)  : G v → (a1,..., ar) and Kq * G}. Of these, the only Folkman number of the form F (2,..., 2; r − 1) which has remained unknown up to this r time is Fv(2, 2, 2, 2, 2; 4). We show here that Fv(2, 2, 2, 2, 2; 4) = 16, which is equivalent to saying that the smallest 6chromatic K4free graph has 16 vertices. We also show that the sole witnesses of the upper bound Fv(2, 2, 2, 2, 2; 4) ≤ 16 are the two Ramsey (4,4)graphs on 16 vertices. 1
Computing the Folkman Number Fv(2, 2, 3; 4)
, 2005
"... We discuss a branch of Ramsey theory concerning vertex Folkman numbers and how computer algorithms have been used to compute a new Folkman number. We write G → (a1,..., ak) v if for every vertex kcoloring of an undirected simple graph G, a monochromatic Ka i is forced in color i ∈ {1,..., k}. The v ..."
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We discuss a branch of Ramsey theory concerning vertex Folkman numbers and how computer algorithms have been used to compute a new Folkman number. We write G → (a1,..., ak) v if for every vertex kcoloring of an undirected simple graph G, a monochromatic Ka i is forced in color i ∈ {1,..., k}. The vertex Folkman number is defined as Fv(a1,..., ak; p) = min{V (G)  : G → (a1,..., ak) v ∧ Kp � ⊆ G}. Folkman showed in 1970 that this number exists for p> max{a1,..., ak}. Let m = 1+ P k i=1 (ai−1) and a = max{a1,..., ak}, then Fv(a1,..., ak; p) = m for p> m, and Fv(a1,..., ak; p) = a + m for p = m. For p < m the situation is more difficult and much less is known. We show here that, for a case of p = m−1, Fv(2, 2, 3; 4) = 14. 1
Uses of Randomness in Computation
, 1994
"... Random number generators are widely used in practical algorithms. Examples include simulation, number theory (primality testing and integer factorization), fault tolerance, routing, cryptography, optimization by simulated annealing, and perfect hashing. Complexity theory usually considers the worst ..."
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Random number generators are widely used in practical algorithms. Examples include simulation, number theory (primality testing and integer factorization), fault tolerance, routing, cryptography, optimization by simulated annealing, and perfect hashing. Complexity theory usually considers the worstcase behaviour of deterministic algorithms, but it can also consider averagecase behaviour if it is assumed that the input data is drawn randomly from a given distribution. Rabin popularised the idea of &quot;probabilistic &quot; algorithms, where randomness is incorporated into the algorithm instead of being assumed in the input data. Yao showed that there is a close connection between the complexity of probabilistic algorithms and the averagecase complexity of deterministic algorithms. We give examples of the uses of randomness in computation, discuss the contributions of Rabin, Yao and others, and mention some open questions.
Recent Results in Computational Ramsey Theory
, 1992
"... Since its conception in 1930, classical and generalized Ramsey theory has been widely studied. Despite this, very little progress has been made, due to the intractability of the problems encountered. In the past year, several new exact values and improved bounds have been discovered for Ramsey nu ..."
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Since its conception in 1930, classical and generalized Ramsey theory has been widely studied. Despite this, very little progress has been made, due to the intractability of the problems encountered. In the past year, several new exact values and improved bounds have been discovered for Ramsey numbers, by using a hybrid approach of clever mathematics and large scale computer search. The validity of this increasingly important method of analysis raises some important questions which must be resolved by mathematicians and computer scientists.
On some open questions for Ramsey . . .
, 2014
"... We discuss some of our favorite open questions about Ramsey numbers and a related problem on edge Folkman numbers. For the classical twocolor Ramsey numbers we first focus on constructive bounds for the difference between consecutive Ramsey numbers. We present the history of progress on the Ramsey ..."
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We discuss some of our favorite open questions about Ramsey numbers and a related problem on edge Folkman numbers. For the classical twocolor Ramsey numbers we first focus on constructive bounds for the difference between consecutive Ramsey numbers. We present the history of progress on the Ramsey number R(5, 5) and discuss the conjecture that it is equal to 43. For the multicolor Ramsey numbers we focus on the growth of Rr(k), in particular for k = 3. Two concrete conjectured cases, R(3, 3, 3, 3) = 51 and R(3, 3, 4) = 30, are discussed in some detail. For Folkman numbers, we present the history, recent developments and potential future progress on Fe(3, 3; 4), defined as the smallest number of vertices in any K4free graph which is not a union of two trianglefree graphs. Although several problems discussed in this paper are concerned with concrete cases, and some involve significant computational approaches, there are interesting and important theoretical questions behind each of them.