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

Andrew Frohmader
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.  ... 
arXiv:0806.4208v1 fatcat:j2jlksr5ffdkrdvszpsqgvwa2u

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

Alexander Razborov
2012 arXiv   pre-print
We identify three 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  ... 
arXiv:1210.4605v1 fatcat:om3smy5auned3ojfxlgm4in73e

Syntactic Separation of Subset Satisfiability Problems

Eric Allender, Martín Farach-Colton, Meng-Tsung Tsai, Michael Wagner
2019 International Workshop on Approximation Algorithms for Combinatorial Optimization  
In this paper, we consider 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] ).  ... 
doi:10.4230/lipics.approx-random.2019.16 dblp:conf/approx/AllenderFT19 fatcat:g67vpjt2fvb4hhzurk3s5fknr4

On the existence of triangulated spheres in 3-graphs, and related problems

V. T. Sós, P. Erdős, W. G. Brown
1973 Periodica Mathematica Hungarica  
The case d = Tc = 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 .  ... 
doi:10.1007/bf02018585 fatcat:n3khwuwbxzc43lcu2fwk6c4t4m

Localized versions of extremal problems [article]

David Malec, Casey Tompkins
2022 arXiv   pre-print
We generalize several classical theorems in extremal combinatorics by replacing 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.  ... 
arXiv:2205.12246v2 fatcat:4nmjfoehvrbfxd2gglz5p2y2gu

On the Fon-der-Flaass Interpretation of Extremal Examples for Turan's (3,4)-problem [article]

Alexander Razborov
2010 arXiv   pre-print
Fon-der-Flaass (1988) presented 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.  ... 
arXiv:1008.4707v1 fatcat:3hhiiogukff6zi7utoxpxfvf7m

Multicolour Turán problems

Peter Keevash, Mike Saks, Benny Sudakov, Jacques Verstraëte
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  ... 
doi:10.1016/j.aam.2003.08.005 fatcat:umtvbqkdm5ga3cxpnf2ic3hrsq

A hypergraph Turán problem with no stability [article]

Xizhi Liu, Dhruv Mubayi
2021 arXiv   pre-print
Indeed, the classical 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.  ... 
arXiv:1911.07969v2 fatcat:2f24ratt4fg3paeub534t5r6ci

On a colored Turán problem of Diwan and Mubayi [article]

Ander Lamaison, Alp Müyesser, Michael Tait
2020 arXiv   pre-print
This 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.  ... 
arXiv:2010.02953v1 fatcat:gejv3wr4bveorhn6nc3bwjwr7m

Solving Turán's Tetrahedron Problem for the ℓ_2-Norm [article]

József Balogh and Felix Christian Clemen and Bernard Lidický
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.  ... 
arXiv:2108.10408v2 fatcat:gggj4gxjd5hxdorxokx6j7uo4y

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

Sounaka Mishra, Shijin Rajakrishnan
2018 arXiv   pre-print
This approximation factor is 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.  ... 
arXiv:1712.02870v3 fatcat:7smt2kc43bal5ce6ojdgfxk73i

Turán and Ramsey problems for alternating multilinear maps [article]

Youming Qiao
2021 arXiv   pre-print
Given s, t∈ℕ, s, t≥ 2, and an alternating bilinear map f:V× V→ U with (V)=s· t^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.  ... 
arXiv:2007.12820v3 fatcat:wf5bx5epn5amfhc37bt2q36rzq

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.  ... 
doi:10.1017/s0963548311000319 fatcat:cka4g6d5drddpiqjnkgbhaevr4

Extremal problems in uniformly dense hypergraphs [article]

Christian Reiher
2019 arXiv   pre-print
Already the case when F is 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.  ... 
arXiv:1901.04027v1 fatcat:rtc73vt26jcxhdnwbh5qtmwoqu

Subgraph Domatic Problem and Writing Capacity of Memory Devises with Restricted State Transitions [article]

Tadashi Wadayama, Taizuke Izumi, Hirotaka Ono
2015 arXiv   pre-print
The goal 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.  ... 
arXiv:1501.04402v1 fatcat:5hdbsso5wrejlamwrild554nxy
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