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

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

arXiv:1902.07693v3
fatcat:2qfonnctpbf3pcx4grizgw5n7m
*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. ...##
###
On the Erdős-Sós conjecture for trees with bounded degree
[article]

2020
*
arXiv
*
pre-print

We prove

arXiv:1906.10219v2
fatcat:qyve2odkrjehjmps6glmfklxbu
*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*. ...##
###
Advances on the Conjecture of Erdős-Sós for spiders
[article]

2017
*
arXiv
*
pre-print

We claim also that

arXiv:1706.03414v1
fatcat:zwaalu5zvndujgfabbsqccnsui
*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. ...##
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On the Erdos-Sos Conjecture for Graphs on n=k+4 Vertices
[article]

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 ′ . ...

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

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. ...

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

2003
*
Combinatorics, probability & computing
*

Acknowledgement We thank an anonymous referee for many worthwhile suggestions and correction of several errors in

doi:10.1017/s096354830300590x
fatcat:gdp2exrux5cirjezdpeupk6b2y
*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. ...##
###
On the Erdős-Gyárfás conjecture in claw-free graphs
[article]

2013
*
arXiv
*
pre-print

Since this

arXiv:1109.5398v3
fatcat:u5vut7kttbar7pakkg7zktj7bu
*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. ...##
###
Open problems of Paul Erd�s in graph theory

1997
*
Journal of Graph Theory
*

There are many survey papers

doi:10.1002/(sici)1097-0118(199705)25:1<3::aid-jgt1>3.0.co;2-r
fatcat:ja46si6w75gvzh2ksrkvblnznm
*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. ...##
###
The Erdos and Campbell-Staton conjectures about square packing
[article]

2005
*
arXiv
*
pre-print

Campbell and Staton, building

arXiv:math/0504341v1
fatcat:siusrdxxyfedndyvw4pccw7jzi
*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. ...##
###
On a conjecture of Erdős concerning primitive sequences
[article]

2017
*
arXiv
*
pre-print

Then, we show (by a counterexample) that

arXiv:1104.3724v2
fatcat:qdopoyhqvzb7ja5yfkabu5gova
*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. ...##
###
On a conjecture of Erdős
[article]

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. ...

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

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

arXiv:1206.3966v3
fatcat:hxyuuqsykbfedlpinm4ivkdyji
*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*...##
###
Results and conjectures related to a conjecture of Erdős concerning primitive sequences
[article]

2017
*
arXiv
*
pre-print

In

arXiv:1709.08708v2
fatcat:staia6gafjf2feyam7zgu7aeum
*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 ...##
###
On a conjecture of Erdös and certain Dirichlet series
[article]

2015
*
arXiv
*
pre-print

If q≡ 3 ( mod 4), Murty and Saradha proved

arXiv:1501.04185v1
fatcat:3u5arboonnfvvo6ee6slljrzde
*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. ...##
###
On the Erd\H{o}s-Gyárfás conjecture in claw-free graphs

2014
*
Discussiones Mathematicae Graph Theory
*

Since this

doi:10.7151/dmgt.1732
fatcat:pjx3u33upnbwjmpq5rncbgsvva
*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. ...
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