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More Constructions for Turán's (3, 4)-Conjecture
[article]

2008
*
arXiv
*
pre-print

*For*

*Turán's*(

*3*,

*4*)-conjecture, in the case

*of*n = 3k+1 vertices, (.5)6^k-1 non-isomorphic complexes are

*constructed*that attain the conjecture. ... In the case

*of*n = 3k+2 vertices, 6^k-1 non-isomorphic complexes are

*constructed*that attain the conjecture. ... Some upper bounds

*for*the k =

*3*, r =

*4*case

*of*

*Problem*1.1 are also known. ...

##
###
On Turan's (3,4)-problem with forbidden configurations
[article]

2012
*
arXiv
*
pre-print

We identify three

arXiv:1210.4605v1
fatcat:om3smy5auned3ojfxlgm4in73e
*3*-graphs on five vertices each missing in all known extremal configurations*for**Turan's*(3,4)-*problem*and prove*Turan's*conjecture*for**3*-graphs that are additionally known not to contain ... any induced copies*of*these*3*-graphs. ... Specifically, we are trying to utilize one*of*the results in [Raz11] that solves*Turan's*(3,4)-*problem**for*the*class**of**3*-graphs resulting, via Fon-der-Flaas interpretation, from*a**class**of*oriented ...##
###
Syntactic Separation of Subset Satisfiability Problems

2019
*
International Workshop on Approximation Algorithms for Combinatorial Optimization
*

In this paper, we consider

doi:10.4230/lipics.approx-random.2019.16
dblp:conf/approx/AllenderFT19
fatcat:g67vpjt2fvb4hhzurk3s5fknr4
*a*syntactically defined*class**of**problems*, and give conditions*for*when*problems*in this*class*require strongly exponential time to approximate to within*a*factor*of*(1 − ε) ... Our*class*includes*a*rich set*of**problems*from additive combinatorics, computational geometry, and graph theory. Our hardness results also match the best known algorithmic results*for*these*problems*. ... In our proofs, when we refer to "*Turán's*Theorem", we refer to the second formulation*of**Turán's*Theorem [53] , that is: Theorem*4*(*Turán's*Theorem [55, 53] ). ...##
###
On the existence of triangulated spheres in 3-graphs, and related problems

1973
*
Periodica Mathematica Hungarica
*

The case d = Tc =

doi:10.1007/bf02018585
fatcat:n3khwuwbxzc43lcu2fwk6c4t4m
*4*is the first open case*of**Turán's**problem*[20] . b . k = 2. The lower bound follows from the G(*3*) (n, [n /*3*]) which has no two triples sharing*a*vertex . ... Thus*for*all*a*, ex ( , n,, W) - < (n -1) 2*3*. b . We*construct**for*every integer ra,*a**3*-graph*A*(30 (n ; n(n -*4*)~*3*) containing no wheels . ...##
###
Localized versions of extremal problems
[article]

2022
*
arXiv
*
pre-print

We generalize several classical theorems in extremal combinatorics by replacing

arXiv:2205.12246v2
fatcat:4nmjfoehvrbfxd2gglz5p2y2gu
*a*global constraint with an inequality which holds*for*all objects in*a*given*class*. ... In particular we obtain generalizations*of**Turán's*theorem, the Erdős-Gallai theorem, the LYM-inequality, the Erdős-Ko-Rado theorem and the Erdős-Szekeres theorem on sequences. ...*Turán's*theorem asserts that*for*all r we have ex(n, K r+1 ) ≤ (r−1)n 2 2r , and equality holds when r divides n and we take*a*complete r-partite graph with*classes**of*equal size. ...##
###
On the Fon-der-Flaass Interpretation of Extremal Examples for Turan's (3,4)-problem
[article]

2010
*
arXiv
*
pre-print

Fon-der-Flaass (1988) presented

arXiv:1008.4707v1
fatcat:3hhiiogukff6zi7utoxpxfvf7m
*a*general*construction*that converts an arbitrary C⃗_4-free orgraph Γ into*a*Turan (3,4)-graph. ... In 1941, Turan conjectured that the edge density*of*any*3*-graph without independent sets on*4*vertices (Turan (3,4)-graph) is >=*4*/9(1-o(1)), and he gave the first example witnessing this bound. ... To the best*of*our knowledge, this is the first result*of*this kind*for**Turán's*(*3*,*4*)-*problem*. ...##
###
Multicolour Turán problems

2004
*
Advances in Applied Mathematics
*

*A*simple k-colouring

*of*

*a*multigraph G is

*a*decomposition

*of*the edge multiset as the sum

*of*k simple graphs, called 'colours'. ...

*Turán's*

*problem*). ...

*For*example,

*a*generalisation

*of*

*Turán's*

*problem*introduced by Erdös in 1963 asks

*for*the largest number

*of*edges in

*a*graph such that every r vertices span at most s edges (the case s = r 2 − 1 being ...

##
###
A hypergraph Turán problem with no stability
[article]

2021
*
arXiv
*
pre-print

Indeed, the classical

arXiv:1911.07969v2
fatcat:2f24ratt4fg3paeub534t5r6ci
*constructions*due to Kostochka imply that the notorious extremal*problem**for*the tetrahedron exhibits this phenomenon assuming*Turán's*conjecture. ...*A*fundamental barrier in extremal hypergraph theory is the presence*of*many near-extremal*constructions*with very different structures. ...*For*ℓ > r ≥*3*, let K r ℓ be the complete r-graph on ℓ vertices. Extending*Turán's*theorem to hypergraphs (i.e. r ≥*3*) is*a*major*problem*. ...##
###
On a colored Turán problem of Diwan and Mubayi
[article]

2020
*
arXiv
*
pre-print

This

arXiv:2010.02953v1
fatcat:gejv3wr4bveorhn6nc3bwjwr7m
*problem*was introduced by Diwan and Mubayi, who conjectured that (except*for**a*few specific exceptions) when H is*a*complete graph on k+1 vertices with any coloring*of*its edges mex(n,H)=ex(n, K_k ... This conjecture generalizes*Turán's*theorem. Diwan and Mubayi also asked*for*an analogue*of*Erdős-Stone-Simonovits theorem in this context. ... See Figure 1*for**a*diagram*of*the*construction*in the case when r =*4*and k =*3*. ...##
###
Solving Turán's Tetrahedron Problem for the ℓ_2-Norm
[article]

2021
*
arXiv
*
pre-print

*Turán's*famous tetrahedron

*problem*is to compute the Turán density

*of*the tetrahedron K_4^

*3*. ... In particular, we prove that the extremal K_4^

*3*-free hypergraphs in ℓ_2-norm have approximately the same structure as one

*of*the conjectured extremal hypergraphs

*for*

*Turán's*conjecture. ... Acknowledgements We thank an anonymous referee

*for*many useful comments and suggestions, in particular

*for*pointing out

*a*shorter proof

*of*Theorem 1.5. ...

##
###
The Complexity of Maximum k-Order Bounded Component Set Problem
[article]

2018
*
arXiv
*
pre-print

This approximation factor is

arXiv:1712.02870v3
fatcat:7smt2kc43bal5ce6ojdgfxk73i
*a*generalization*of**Turán's*approximation factor*for*Max-IS. ... We generalize*Turán's*greedy algorithm*for*Max-IS and prove that it approximates Max-k-OBCS within*a*factor*of*(2k - 1)d + k, where d is the average degree*of*the input graph G. ... This approximation factor is*a*generalization*of**Turán's*bound*for*MAX-IS. ...##
###
Turán and Ramsey problems for alternating multilinear maps
[article]

2021
*
arXiv
*
pre-print

Given s, t∈ℕ, s, t≥ 2, and an alternating bilinear map f:V× V→ U with (V)=s· t^

arXiv:2007.12820v3
fatcat:wf5bx5epn5amfhc37bt2q36rzq
*4*, we show that there exists either*a*dimension-s subspace W≤ V such that (f(W, W))=0, or*a*dimension-t subspace W≤ V such ... Our main result is*a*Ramsey theorem*for*alternating bilinear maps. ... Recall that*a*variety*of*groups, C, is the*class**of*all groups satisfying*a*set*of*laws. Examples include abelian groups, nilpotent groups*of**class*c, and solvable groups*of**class*c. ...##
###
Daisies and Other Turán Problems

2011
*
Combinatorics, probability & computing
*

*A*daisy, or r-daisy, is

*a*certain r-uniform hypergraph consisting

*of*six sets: given an (r − 2)-set P and

*a*

*4*-set Q disjoint from P , the daisy on (P, Q) consists

*of*the r-sets

*A*with P ⊂

*A*⊂ P ∪ Q. ...

*For*

*a*set X, we write X (r)

*for*the set

*of*all r-element subsets

*of*X. An r-graph (or r-uniform hypergraph) on X is

*a*subset

*of*X (r) . ... /r r , as the complete r-partite r-graph does not contain

*a*daisy.

*For*r = 2,

*a*daisy is precisely

*a*K

*4*, and so

*Turán's*theorem tells us that π(D 2 ) = 2/

*3*. ...

##
###
Extremal problems in uniformly dense hypergraphs
[article]

2019
*
arXiv
*
pre-print

Already the case when F is

arXiv:1901.04027v1
fatcat:rtc73vt26jcxhdnwbh5qtmwoqu
*a*k-uniform hypergraph with three edges on k+1 vertices is still wide open even*for*k=*3*. ...*For**a*k-uniform hypergraph F let ex(n,F) be the maximum number*of*edges*of**a*k-uniform n-vertex hypergraph H which contains no copy*of*F. ...*For*numerous reasons going much beyond our collaboration [25] [26] [27] [28] [29] my indebtedness to Vojtěch Rödl and Mathias Schacht is extremely great. ...##
###
Subgraph Domatic Problem and Writing Capacity of Memory Devises with Restricted State Transitions
[article]

2015
*
arXiv
*
pre-print

The goal

arXiv:1501.04402v1
fatcat:5hdbsso5wrejlamwrild554nxy
*of**a*subDP*problem*is to find*a*valid coloring with the largest number*of*colors*for**a*subgraph*of**a*given directed graph. ...*A*code design*problem**for*memory devises with restricted state transitions is formulated as*a*combinatorial optimization*problem*that is called*a*subgraph domatic partition (subDP)*problem*. ... ACKNOWLEDGEMENT The first author would like to express his sincere appreciation to Hiroshi Kamabe and Kees Immink*for*their*constructive*comments*for*an earlier version*of*this work. ...
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