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### The covering lemma for K

Tony Dodd, Ronald Jensen
1982 Annals of Mathematical Logic
t(~ the covering lemma for L.  ...  to pre,~ent these last two results in a .3eparate paper "The covering lemma for L[U]".  ...  The result is true even without (b) but ~he proof is more involved, and the stated result suffices for the covering lemma. Section 3 contains some messy technicalities about well foundedness.  ...

### Periodic orbits of continuous mappings of the circle

Louis Block
1980 Transactions of the American Mathematical Society
It is shown that if 1 G P(f) and n G P(f) for some odd positive integer n then for every integer m > n, m G P(f).  ...  Let / be a continuous map of the circle into itself and let P(f) denote the set of positive integers n such that/has a periodic point of period n.  ...  Suppose that for some positive integery with s < / < ^ -1, L /-covers T". Then the conclusion of the theorem holds by Lemma 7 (with k = 3, Mx = [e,px], M2 = Tr_, and M3 = IJ).  ...

### Periodic Orbits of Continuous Mappings of the Circle

Louis Block
1980 Transactions of the American Mathematical Society
It is shown that if 1 G P(f) and n G P(f) for some odd positive integer n then for every integer m > n, m G P(f).  ...  Let / be a continuous map of the circle into itself and let P(f) denote the set of positive integers n such that/has a periodic point of period n.  ...  Suppose that for some positive integery with s < / < ^ -1, L /-covers T". Then the conclusion of the theorem holds by Lemma 7 (with k = 3, Mx = [e,px], M2 = Tr_, and M3 = IJ).  ...

### Upper bounds on the sizes of variable strength covering arrays using the Lovász local lemma [article]

Lucia Moura, Sebastian Raaphorst, Brett Stevens
2019 arXiv   pre-print
is the special case where coverage is required for all sets of columns of a fixed size t, its strength.  ...  The conclusions are dependent on the class of hypergraph, and we discuss specific characteristics of the hypergraphs which are more amenable to using different versions of the Lovász local lemma.  ...  The authors would like to thank an anonymous reviewer for various suggestions that improved the presentation of this paper.  ...

### On coverings

Johanan Schonheim
1964 Pacific Journal of Mathematics
The existence of an admissible covering M(k -1, 1,1, n) is obvious. Suppose now that the lemma is proved for I -l 0 and let I -l 0 + 1.  ...  coverings which are not tactical configurations* The fact that there exist admissible coverings which are not tactical configurations will be shown in Corollaries 1, 2 and 3 to Lemma 3, but for the purpose  ...

### Quasi-Differential Posets and Cover Functions of Distributive Lattices

Jonathan David Farley
2000 Journal of combinatorial theory. Series A
A function f : N 0 Ä N 0 is a cover function for L if every element with n lower covers has f (n) upper covers.  ...  In this paper, all finitary distributive lattices with non-decreasing cover functions are characterized. A 1975 conjecture of Richard P. Stanley is thereby settled. Academic Press  ...  Example 3 . 2 . 32 For k # N, the function f (n)=k+n (n # N 0 ) is a cover function for Y k . (See Lemma 4.8.) Example 3.3.  ...

### On Ryser's conjecture

P. E. Haxell, A. D. Scott
2012 Electronic Journal of Combinatorics
Motivated by an old problem known as Ryser's Conjecture, we prove that for $r=4$ and $r=5$, there exists $\epsilon>0$ such that every $r$-partite $r$-uniform hypergraph $\cal H$ has a cover of size at  ...  most $(r-\epsilon)\nu(\cal H)$, where $\nu(\cal H)$ denotes the size of a largest matching in $\cal H$.  ...  For each x k that exists and lies in a cover of size two of the C l it is in, set z k to be the other vertex of the cover. Note that z k is unique by Lemma 2.3. Define z k similarly for each x k .  ...

### Covering Numbers in Linear Algebra [article]

Pete L. Clark
2012 arXiv   pre-print
We compute the minimal cardinality of a covering (resp. an irredundant covering) of a vector space over an arbitrary field by proper linear subspaces.  ...  Analogues for affine linear subspaces are also given.  ...  Lemma 2. For any field K, the unique linear covering of K 2 is the set of all lines through the origin, of cardinality #K + 1. It is an irredundant covering. Proof.  ...

### Combinatorial Theorems on the Simplotope that Generalize Results on the Simplex and Cube

Robert M. Freund
1986 Mathematics of Operations Research
Each combinatorial theorem also implies set covering lemmas on the simplotope, the simplex, and the cube, including the Generalized Covering lemma, the Knaster-Kuratowski-Mazurkiewicz lemma, and a lemma  ...  This paper presents three combinatorial theorems on the simplotope, and shows how each translates into some known and new results on the simplex and cube, including various forms of Sperner's lemma.  ...  For the details of the proof, Cn I vk = 1, k = 1,..., n. Then there exists j e {1,..., nI such that Di n D j # I. Covering lemma 3 on the Cube: Let D ,..., Covering lemma 2 on the Cube, n = 2.  ...

### Small cycle covers of 3-connected cubic graphs

Fan Yang, Xiangwen Li
2011 Discrete Mathematics
It is proved in this paper that G admits a cycle cover of size at most ⌈n/6⌉ if and only if G is not one of the five specified graphs.  ...  Acknowledgements The authors would like to thank the referees for the valuable comments and suggestions which improved the representation.  ...  The second author is partially supported by the Natural Science Foundation of China (10571071), the Research Funds of CCNU09Y01018, and Hubei Key Laboratory of Mathematical Sciences of China.  ...

### Covering Numbers in Linear Algebra

Pete L. Clark
2012 The American mathematical monthly
We compute the minimal cardinality of a covering (resp. an irredundant covering) of a vector space over an arbitrary field by proper linear subspaces.  ...  Analogues for affine linear subspaces are also given.  ...  Lemma 2 . 2 For any field K, the unique linear covering of K 2 is the set of all lines through the origin, of cardinality #K + 1. It is an irredundant covering.Proof.  ...

### Subdirect products and covering groups by subgroups

2013 International Journal of Algebra
A cover for a group is a collection of proper subgroups whose union is the whole group.  ...  A cover C for a group G is called a C n -cover whenever C is an irredundant maximal core-free n-cover for G and in this case we say that G is a C n -group.  ...  A set C of proper subgroups of G is called a cover for G if its set-theoretic union is equal to G. If the size of C is n, we call C an n-cover for the group G.  ...

### On uniform approximation of harmonic functions

M. Ya. Mazalov
2012 St. Petersburg Mathematical Journal
The result is an approximation theorem for an individual function under the condition that, on the complement to the compact set, the harmonic capacity is "homogeneous" in a sense.  ...  The proof involves a refinement of Vitushkin's localization method. §0. Introduction Let X ⊂ R 3 be a compact set, X • the interior of X, Δ the Laplace operator in R 3 .  ...  )) (see Lemma 1.5) that corresponds to the new Cover(ν); furthermore, k 7 depends only on k 0 in (0.2) and on k 2 in Lemma 1.5.  ...

### Volumes of balls in Riemannian manifolds and Uryson width [article]

Larry Guth
2015 arXiv   pre-print
If (M^n, g) is a closed Riemannian manifold where every unit ball has volume at most ϵ_n (a sufficiently small constant), then the (n-1)-dimensional Uryson width of (M^n, g) is at most 1.  ...  Our map φ (k−1) will be R δ(k) •φ (k) for a well-chosen δ(k). Notice that for any δ, the map R δ sends each open face F intoF . By Lemma 1.1, R δ •φ (k) is subordinate to the cover.  ...  In particular, we get the following lemma. Lemma 1.2. Suppose that φ (k) : X → N (k) is a map from X to the k-skeleton of N subordinate to the cover.  ...

### Local colourings and monochromatic partitions in complete bipartite graphs

Richard Lang, Maya Stein
2017 European journal of combinatorics (Print)
We show that for any 2-local colouring of the edges of a complete bipartite graph, its vertices can be covered with at most 3 disjoint monochromatic paths.  ...  And, we can cover almost all vertices of any complete or complete bipartite r-locally coloured graph with O(r 2 ) disjoint monochromatic cycles.  ...  In total, we used r(r + 1)/2 + r 2 + 1 ≤ 2r 2 disjoint monochromatic cycles to cover all of K n . This proves Theorem 1.1(a).  ...
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