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A note on the Brown--Erdős--Sós conjecture in groups [article]

Jason Long
2019 arXiv   pre-print
We confirm that for all t we can find a collection of t triples spanning at most t+3 vertices, resolving the Brown--Erdős--Sós conjecture in this context.  ...  The proof applies well-known arithmetic results including the multidimensional versions of Szemerédi's theorem and the density Hales--Jewett theorem.  ...  Acknowledgements The author would like to thank Tim Gowers for several helpful comments.  ... 
arXiv:1902.07693v3 fatcat:2qfonnctpbf3pcx4grizgw5n7m

On the Erdős-Sós conjecture for trees with bounded degree [article]

Guido Besomi, Matías Pavez-Signé, Maya Stein
2020 arXiv   pre-print
We prove the Erd\H os--S\'os conjecture for trees with bounded maximum degree and large dense host graphs.  ...  As a corollary, we obtain an upper bound on the multicolour Ramsey number of large trees whose maximum degree is bounded by a constant.  ...  Erdős and Graham also observed that the upper bound in (1) would follow from the Erdős-Sós conjecture.  ... 
arXiv:1906.10219v2 fatcat:qyve2odkrjehjmps6glmfklxbu

Advances on the Conjecture of Erdős-Sós for spiders [article]

Camino Balbuena and Mucuy-Kak Guevara and José R. Portillo and Pedro Reyes
2017 arXiv   pre-print
We claim also that the condition of 2-connection is not needed, but the proof is very long and it is not included in this document.  ...  Introduction The Erdős-Sós conjecture [2] says that a graph G on n vertices and number of edges e(G) > n(k − 1)/2 contains all trees of size k.  ...  By g(n, k) Erdős and Gallai [3] denoted the maximum number of edges of a graph G on n vertices containing no cycles with more than k edges.  ... 
arXiv:1706.03414v1 fatcat:zwaalu5zvndujgfabbsqccnsui

On the Erdos-Sos Conjecture for Graphs on n=k+4 Vertices [article]

Long-Tu Yuan, Xiao-Dong Zhang
2014 arXiv   pre-print
The Erdős-Sós Conjecture states that if G is a simple graph of order n with average degree more than k-2, then G contains every tree of order k.  ...  In this paper, we prove that Erdős-Sós Conjecture is true for n=k+4.  ...  So avedeg(G ′ ) > (k 2 − 2k − 8)/(k + 2) = k − 4 and | V (T ′ ) |≤ k − 2. By the induction hypothesis, T ′ ⊆ G ′ .  ... 
arXiv:1403.5430v1 fatcat:c7dzkzyj3jgf7gb27xnd2olmzi

On the Erdős-Sós Conjecture for graphs on n = k + 4 vertices

Long-Tu Yuan, Xiao-Dong Zhang
2016 Ars Mathematica Contemporanea  
The Erdős-Sós Conjecture states that if G is a simple graph of order n with average degree more than k − 2, then G contains every tree of order k.  ...  In this paper, we prove that Erdős-Sós Conjecture is true for n = k + 4.  ...  Based on the above result, Later Erdős and Sós proposed the following well known conjecture (for example, see [5] ). Conjecture 1.2. Let G be a graph with avedeg(G) > k − 2.  ... 
doi:10.26493/1855-3974.905.cb4 fatcat:mfbvadkgzzer3kun642tlcdtbm

On the Erdős–Simonovits–Sós Conjecture about the Anti-Ramsey Number of a Cycle

TAO JIANG, DOUGLAS B. WEST
2003 Combinatorics, probability & computing  
Acknowledgement We thank an anonymous referee for many worthwhile suggestions and correction of several errors in the proof of the second theorem.  ...  Concluding remarks Our results suggest several approaches to proving the full Erdős-Simonovits-Sós conjecture.  ...  Erdős, Simonovits and Sós [6] initiated the study of f(n, C k ). For fixed k with k 3, they conjectured that f(n, C k ) = k − 2 2 + 1 k − 1 n + O (1) and proved this for k = 3.  ... 
doi:10.1017/s096354830300590x fatcat:gdp2exrux5cirjezdpeupk6b2y

On the Erdős-Gyárfás conjecture in claw-free graphs [article]

Pouria Salehi Nowbandegani, Hossein Esfandiari, Mohammad Hassan Shirdareh Haghighi, Khodakhast Bibak
2013 arXiv   pre-print
Since this conjecture has proven to be far from reach, Hobbs asked if the Erdős-Gyárfás conjecture holds in claw-free graphs.  ...  The Erdős-Gyárfás conjecture states that every graph with minimum degree at least three has a cycle whose length is a power of 2.  ...  In fact, in the next section, we obtain two results on the Erdős-Gyárfás conjecture in claw-free graphs with δ ≥ 3, and δ ≥ 4.  ... 
arXiv:1109.5398v3 fatcat:u5vut7kttbar7pakkg7zktj7bu

Open problems of Paul Erd�s in graph theory

F. R. K. Chung
1997 Journal of Graph Theory  
There are many survey papers on the impact of Paul's work, e.g., see those in the books: "A Tribute to Paul Erdős" [84] , "Combinatorics, Paul Erdős is Eighty", Volumes 1 and 2 [83], and "The Mathematics  ...  Through the problems, the legacy of Paul Erdős continues (particularly if solving one of these problems results in creating three new problems, for example.)  ...  Acknowledgement The author wishes to thank Paul Seymour for suggesting writing this paper.  ... 
doi:10.1002/(sici)1097-0118(199705)25:1<3::aid-jgt1>3.0.co;2-r fatcat:ja46si6w75gvzh2ksrkvblnznm

The Erdos and Campbell-Staton conjectures about square packing [article]

Iwan Praton
2005 arXiv   pre-print
Campbell and Staton, building on a question of Erdos, conjectured that f(k^2+2c+1)=k+c/k, where c is any integer and k\geq |c|.  ...  We show that if this conjecture is true for one value of c, then it is true for all values of c.  ...  In this short note we show that the Erdös conjecture implies the Campbell-Staton conjectures. Of course, we should describe the conjectures first. Let c be any integer.  ... 
arXiv:math/0504341v1 fatcat:siusrdxxyfedndyvw4pccw7jzi

On a conjecture of Erdős concerning primitive sequences [article]

Bakir Farhi
2017 arXiv   pre-print
Then, we show (by a counterexample) that the analogue of a conjecture of Erdős, for those series, is false.  ...  In this note, we propose a conjecture stating that some series involving primitive sequences are convergent.  ...  Now, we will prove that the analogue of the above conjecture of Erdős for the series considered in Conjecture 1 is false.  ... 
arXiv:1104.3724v2 fatcat:qdopoyhqvzb7ja5yfkabu5gova

On a conjecture of Erdős [article]

Yong-Gao Chen, Yuchen Ding
2022 arXiv   pre-print
Erdős conjectured that if c is an arbitrarily given constant, x is sufficiently large and a_1,... , a_t are positive integers with a_1log x, then there exists an integer n so that the number of solutions  ...  In this note, we confirm this old conjecture of Erdős.  ...  Acknowledgments The first author was supported by the National Natural Science Foundation of China, Grant No. 12171243.  ... 
arXiv:2201.10727v1 fatcat:ukkaij6lpffftfahokcmuoemiu

The inverse Erdos-Heilbronn Problem for restricted set addition in finite groups [article]

Suren Jayasuriya, Steve Reich, Jeffrey Paul Wheeler
2013 arXiv   pre-print
We prove a slight extension to an inverse theorem of Dias da Silva-Hamidoune in Z/pZ, and we present a counterexample to an open conjecture concerning the inverse Erdos-Heilbronn problem in nonabelian  ...  We provide a survey of results concerning both the direct and inverse problems to the Cauchy-Davenport theorem and Erdos-Heilbronn problem in Additive Combinatorics.  ...  [9] early develop the Erdős-Heilbronn conjecture Erdős, Paul 1960's Heilbronn, Hans 1963 states EHP at Number Theory conference Erdős, Paul [11] 1964 paper on sumsets of congruence classes Erdős  ... 
arXiv:1206.3966v3 fatcat:hxyuuqsykbfedlpinm4ivkdyji

Results and conjectures related to a conjecture of Erdős concerning primitive sequences [article]

Bakir Farhi
2017 arXiv   pre-print
In the first part of the paper, we give two significant conjectures which are equivalent to that of Erdős and in the second one, we study the series of the form ∑_a ∈A1/a ( a + x), where x is a fixed non-negative  ...  Besides, he conjectured that the upper bound of the preceding sums is reached when A is the sequence of the prime numbers. The purpose of this paper is to study the Erdős conjecture.  ...  A conjectural answer of this question is proposed by the following conjecture, generalizing the Erdős one while remaining more vague: (where A runs on the set of all primitive sequences different from  ... 
arXiv:1709.08708v2 fatcat:staia6gafjf2feyam7zgu7aeum

On a conjecture of Erdös and certain Dirichlet series [article]

Tapas Chatterjee, M. Ram Murty
2015 arXiv   pre-print
If q≡ 3 ( mod 4), Murty and Saradha proved the conjecture. We show that this conjecture is true for 82% of the remaining integers q≡ 1 ( mod 4).  ...  We thank Michael Roth for his help on using the Maple language as well as Sanoli Gun, Purusottam Rath and Ekata Saha for their comments on an earlier version of this paper.  ...  We also thank the referee for helpful comments that improved the quality of the paper.  ... 
arXiv:1501.04185v1 fatcat:3u5arboonnfvvo6ee6slljrzde

On the Erd\H{o}s-Gyárfás conjecture in claw-free graphs

Khodakhast Bibak, Hossein Esfandiari, Pouria Salehi Nowbandegani, Mohammad Hassan Shirdareh Haghighi
2014 Discussiones Mathematicae Graph Theory  
Since this conjecture has proven to be far from reach, Hobbs asked if the Erdős-Gyárfás conjecture holds in claw-free graphs.  ...  The Erdős-Gyárfás conjecture states that every graph with minimum degree at least three has a cycle whose length is a power of 2.  ...  Acknowledgments The authors would like to thank anonymous referees for helpful mathematical and grammatical comments.  ... 
doi:10.7151/dmgt.1732 fatcat:pjx3u33upnbwjmpq5rncbgsvva
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