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Sum-free sets and ramsey numbers II

D. Hanson, J. Hanson
1977 Discrete Mathematics  
Rweived 25 Octciixx i976 Revised IO Januq '977 In this note we obtain a new lower hound for the Ramsey number R(S,6). The method is computational and the bound obtained is R(5.6) a _, 1.  ...  F (II) For any krt k,*3, there exists two partitionings of the integers g 1,2,..., f(k,, k2) such that in one partitioning 1 E C, and in the ather 1 E Cz. ;' ) 2% If the fint conjecture is true th e  ...  ., m can be partitioned into two classes C, and Cz in such a way that C, does not contain a solution to a system of equation (Sk,) and C2 does not contain a solution to a system of equations (S,,) where  ... 
doi:10.1016/0012-365x(77)90068-1 fatcat:pzhbqru7qfgapkmeu5vrf5hjg4

401 and beyond: improved bounds and algorithms for the Ramsey algebra search [article]

Jeremy F. Alm
2017 arXiv   pre-print
In this paper, we discuss an improvement of an algorithm to search for primes p and coset-partitions of Z/pZ* that yield Ramsey algebras over Z/pZ.  ...  We also prove an upper bound on the modulus p in terms of the number of cosets.  ...  The first idea is that if a set is "unstructured" with respect to addition, then it should contain a solution to x + y = z, and hence not be sum-free.  ... 
arXiv:1609.01817v2 fatcat:3urhfqluyffvblh2hyvlkywkoi

Purely Finitely Additive Measures are Non-Constructible Objects

Luc Lauwers
2010 Social Science Research Network  
The existence of a purely finitely additive measure cannot be proved in Zermelo-Frankel set theory if the use of the Axiom of Choice is disallowed.  ...  Introduction A finitely additive measure µ on N assigns to each subset of N a nonnegative real number and assigns to the union of disjoint sets the sum of their numbers.  ...  Furthermore, to each 3 AC is (i) consistent and (ii) independent: (i) AC can be added to the Zermelo-Fraenkel axioms of set theory (ZF) without yielding a contradiction, and (ii) AC is not a theorem of  ... 
doi:10.2139/ssrn.1604383 fatcat:awwtxtdeafh3bbqufmmad3gozy

Iterated combinatorial density theorems

P Frankl, R.L Graham, V Rödl
1990 Journal of combinatorial theory. Series A  
Omitting all elements which would cause a violation of (ii) or (iii) means omitting one set of lower density zero and a finite number of sets of upper density zero.  ...  FINITE UNIONS AND FINITE SUMS In this section we will first give new proofs for the "finite unions" and "finite sums" theorems: Let us first recall these two important Ramsey-type theorems (see [GRS]  ... 
doi:10.1016/0097-3165(90)90008-k fatcat:c2gqaid3yvabzozjtdtulj3s7u

Page 555 of Mathematical Reviews Vol. 47, Issue 3 [page]

1974 Mathematical Reviews  
A finite set B is called a sum-set if B=S(A) for some A. A characterization of sum-sets is given and various statements are proved about the sets N(B) = {n|(QA)(|A| =n & S(A) = B)}. E. C.  ...  Hajnal [ibid. 7 (1960), 187-191; MR 226718] proved that every %°™”1-function f admits arbitrarily large finite f-free sets, and the author shows that (ii) assuming the continuum hypothesis, there is an  ... 

The chromatic gap and its extremes [article]

András Gyárfás, András Sebõ, Nicolas Trotignon
2011 arXiv   pre-print
While computational problems related to the chromatic gap are hopeless, an interplay between Ramsey theory and matching theory leads to a simple and (almost) exact formula for (n) in terms of Ramsey numbers  ...  The chromatic gap is the difference between the chromatic number and the clique number of a graph. Here we investigate (n), the maximum chromatic gap over graphs on n vertices.  ...  Acknowledgment: Thanks to Zoli Füredi for calling our attention to [4] (and its references).  ... 
arXiv:1108.3444v1 fatcat:itlhk5esbzbynpc4hbwja34tt4

Independent Sets in Hypergraphs and Ramsey Properties of Graphs and the Integers

Robert Hancock, Katherine Staden, Andrew Treglown
2019 SIAM Journal on Discrete Mathematics  
The authors are grateful to Victor Falgas-Ravry for a helpful conversation on [21] , to Hong Liu for a helpful conversation on the graph Ramsey problems considered in this paper, and to Yury Person for  ...  The authors are also grateful to the reviewers for their helpful and careful reviews.  ...  [m] into r sets which are sum-free modulo m + 1.  ... 
doi:10.1137/18m119642x fatcat:ad77ckfgrncrne2dmb3fmtawny

Sum-free sets and Ramsey numbers

D. Hanson
1976 Discrete Mathematics  
in this note we obtain new tower bounds for the Ramsey numbers R(5,S) and R(5,6). The methrld is based on computational results of partitioning the integers into sum-free sets.  ...  WC obtain R(S, 5) > 42 and R(5,6) 2 53.  ...  In [ 1 J we defined the following function which generalizes the notion of sum-free sets: let k 3 3 and fk (!  ... 
doi:10.1016/0012-365x(76)90005-4 fatcat:6v6mhqdfcbadjhk57bcpgolajy

Page 5582 of Mathematical Reviews Vol. , Issue 94j [page]

1994 Mathematical Reviews  
Mynhardt (3-VCTR; Victoria, BC) 94j:05088 05C55 05C70 Tuza, Zsolt (H-AOS-C; Budapest) Monochromatic coverings and tree Ramsey numbers.  ...  The irredundant Ramsey number s(m,n) is then the smallest order of a complete graph in any red-blue colouring of which there is either an m-element set irredundant in the red graph or an n-element set  ... 

The chromatic gap and its extremes

András Gyárfás, András Sebő, Nicolas Trotignon
2012 Journal of combinatorial theory. Series B (Print)  
of the Ramsey-graphs themselves, and leading possibly to new bounds on Ramsey-numbers.  ...  Our study provides some new properties of Ramsey-graphs themselves: it shows that triangle-free Ramsey graphs have high matchability and connectivity properties, therewith providing some new properties  ...  Acknowledgement: The manuscript [4] proves a reformultation of the main content of Theorem 4.1 using our Ramsey-related methods.  ... 
doi:10.1016/j.jctb.2012.06.001 fatcat:d2jwlcmuunejbea34ri3f7boqi

Page 480 of Mathematical Reviews Vol. , Issue 83b [page]

1983 Mathematical Reviews  
minimal natural number 7 such that there exists a ( p,,---,p,)-Ramsey graph G with n vertices and clique number <q.  ...  The symbol R(p,,---,p,) denotes the Ramsey number, i.e., the minimal natural number n such that the complete graph with n vertices is a (p,,°--,p,)-Ramsey graph, and the symbol N(p),°**,P,3q) denotes the  ... 

Page 1720 of Mathematical Reviews Vol. 50, Issue 6 [page]

1975 Mathematical Reviews  
Street and E. G, Whitehead, Jr., Locally maximal sum-free sets (pp. 1405-1409); L. Suranyi, On line critical graphs (pp. 1411-1444); B. Toft, On colour-critical hypergraphs (pp. 1445-1457); W. T.  ...  Yackel, Inequalities and asymptotic bounds for Ramsey numbers. II (pp. 1537-1545); Problems (contributed by J. E. Baumgartner, D. E. Daykin, I. Gutman, D. J. Kleitman, J. Pelikan, M. D. Plummer, R.  ... 

Independent sets in hypergraphs and Ramsey properties of graphs and the integers [article]

Robert Hancock, Katherine Staden, Andrew Treglown
2018 arXiv   pre-print
(ii) The above result also resolves a general subcase of the asymmetric random Ramsey conjecture of Kohayakawa and Kreuter. (iii) All of the above results in fact hold for uniform hypergraphs.  ...  In this paper we use the container method to give relatively short and elementary proofs of a number of results concerning Ramsey (and Turán) properties of (hyper)graphs and the integers.  ...  Acknowledgements The authors are grateful to Victor Falgas-Ravry for a helpful conversation on [21] , to Hong Liu for a helpful conversation on the graph Ramsey problems considered in this paper, and  ... 
arXiv:1701.04754v4 fatcat:i25y2zu76vcmrndaadsqn26wpa

The ramsey number N(3, 3, 3, 3; 2)

Earl Glen Whitehead
1973 Discrete Mathematics  
A partition of the nonzero elements of the finite abelian group Z/72 X Z/72 into four sum-free sets shows that N (3,3,3,3; 2) > 49.  ...  To each symmetric sum-free set of our partition we associate a disI:inct color.  ...  In this paper we. improve both the lower and upper bounds on this Ramsey number. Definition 0.1. Let G be a group with operation +.  ... 
doi:10.1016/0012-365x(73)90174-x fatcat:ct4aisfpqjdixneb7czxq53klq

Page 5874 of Mathematical Reviews Vol. , Issue 93k [page]

1993 Mathematical Reviews  
II (637-668); W. F. Smyth, Sum graphs of small sum number (669-678); J. Spencer [Joel H. Spencer], Sparse random graphs: a continuum of complete theories (679- 690); G.  ...  Erdés [Paul Erdés] and R. J. Faudree, Size Ramsey functions (219-238); P. Erdés [Paul Erdés], E. Gyéri and M. Simonovits, How many edges should be deleted to make a triangle-free graph bipartite?  ... 
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