Szemerédi’s regularity lemma asserts that every graph can be decomposed into relatively few rando... more Szemerédi’s regularity lemma asserts that every graph can be decomposed into relatively few random-like subgraphs. This random-like behavior enables one to find and enumerate subgraphs of a given isomorphism type, yielding the so-called counting lemma for graphs. The combined application of these two lemmas is known as the regularity method for graphs and has proved useful in graph theory, combinatorial
Bulletin of the American Mathematical Society, 1977
A generalization of the Ramsey theorem is stated. This solves a problem of P. Erdös and others. T... more A generalization of the Ramsey theorem is stated. This solves a problem of P. Erdös and others. The result has recent applications in the theory of ultrafilters and model theory.
A super (d; )-regular graph on 2n vertices is a bipartite graph on the classes of vertices V1 and... more A super (d; )-regular graph on 2n vertices is a bipartite graph on the classes of vertices V1 and V2, wherejV1j =jV2j = n, in which the minimum degree and the maximum degree are between (d )n and (d + )n, and for every U V1;W V2 withjUj
Let $\R$ be the set of all finite graphs $G$ with the Ramsey property that every coloring of the ... more Let $\R$ be the set of all finite graphs $G$ with the Ramsey property that every coloring of the edges of $G$ by two colors yields a monochromatic triangle. In this paper we establish a sharp threshold for random graphs with this property. Let $G(n,p)$ be the random graph on $n$ vertices with edge probability $p$. We prove that there
A short proof of the following result of Brown and Buhler is given: For any ϵ > 0 there exists... more A short proof of the following result of Brown and Buhler is given: For any ϵ > 0 there exists n0 = n0(ϵ) such that if A is an abelian group of odd order |A| > n0 and B ⊆ A with |B| > ϵ|A|, then B must contain three distinct elements x, y, z satisfying x + y = 2z.
There are many “Ramsey theorems”, the most famous of which are Schur’s theorem (see Über die Kong... more There are many “Ramsey theorems”, the most famous of which are Schur’s theorem (see Über die Kongruenz x m +y m ≡z m (modp), Jahresber. Dtsch. Math.-Ver. 25, 114-116 (1916)), van der Waerden’s theorem (see B. L. van der Waerden, Beweis einer Baudetschen Vermutung, Nieuw Arch. Wisk. 15, 212-216 (1927)), and Ramsey’s theorem (see F. P. Ramsey, On a problem in formal logic, Proc. London Math. Soc. (2) 30, 264-285 (1930)). These theorems all all deal with partitions of the infinite set of natural numbers. In the present paper the authors are concerned with theorems of a similar kind, except that they now deal with partitions of a finite set of natural numbers {1,2,...,N}.
Let H1, H2, . . ., H k+1 be a sequence of k + 1 finite, undirected, simple graphs. The (multicolo... more Let H1, H2, . . ., H k+1 be a sequence of k + 1 finite, undirected, simple graphs. The (multicolored) Ramsey number r(H1, H2, . . . , H k+1 ) is the minimum integer r such that in every edge-coloring of the complete graph on r vertices by k+1 colors, there is a monochromatic copy of Hi in color i for some 1 ≤ i ≤ k + 1. We describe a general technique that supplies tight lower bounds for several numbers r(H1, H2, . . . , H k+1 ) when k ≥ 2, and the last graph H k+1 is the complete graph Km on m vertices. This technique enables us to determine the asymptotic behaviour of these numbers, up to a polylogarithmic factor, in various cases. In particular we show that r(K3, K3, Km) = Θ(m 3 poly log m), thus solving (in a strong form) a conjecture of Erdős and Sós raised in 1979. Another special case of our result implies that r(C4, C4, Km) = Θ(m 2 poly log m) and that r(C4, C4, C4, Km) = Θ(m 2 / log 2 m). The proofs combine combinatorial and probabilistic arguments with spectral techniques and certain estimates of character sums.
We develop a complexity theory based on the concept of the graph instead of the Boolean function.... more We develop a complexity theory based on the concept of the graph instead of the Boolean function. We show its relation to the Boolean complexity and prove some lower bounds to the complexity of explicitly given graphs.
A random construction gives new examples of simple hypergraphs with high chromatic number that ha... more A random construction gives new examples of simple hypergraphs with high chromatic number that have few edges and/or low maximum degree. In particular, for every integers k ≥ 2, r ≥ 2, and g ≥ 3, there exist r-uniform non-k-colorable hypergraphs of girth at least g with maximum degree at most r k r−1 ln k . This is only 4r 2 ln k times greater than the lower bound by Erdős and Lovász (Colloquia Math Soc János Bolyai 10 (1973), 609-627) that holds for all hypergraphs (without restrictions on girth).
We show that for any k, m, p, c, if G is a K k -free graph on N then there is an independent set ... more We show that for any k, m, p, c, if G is a K k -free graph on N then there is an independent set of vertices in G that contains an (m, p, c)-set. Hence if G is a K k -free graph on N, then one can solve any partition regular system of equations in an independent set. This is a common generalization of partition regularity theorems of Rado (who characterized systems of linear equations Ax = 0 a solution of which can be found monochromatic under any finite coloring of N) and Deuber (who provided another characterization in terms of (m, p, c)-sets and a partition theorem for them), and of Ramsey's theorem itself.
For a fixed 3-uniform hypergraph F, call a hypergraph F-free if it contains no subhypergraph isom... more For a fixed 3-uniform hypergraph F, call a hypergraph F-free if it contains no subhypergraph isomorphic to F. Let ex(n, F) denote the size of a largest F-free hypergraph G ⊆ [n] 3 . Let F n (F) denote the number of distinct labelled F-free G ⊆ [n] 3 . We show that F n (F) = 2 ex(n,F )+o(n 3 ) , and discuss related problems.
Let r 2 be an integer. The real number α ∈ [0, 1] is a jump for r if there exists c > 0 such that... more Let r 2 be an integer. The real number α ∈ [0, 1] is a jump for r if there exists c > 0 such that for every positive and every integer m r, every r-uniform graph with n > n 0 ( , m) vertices and at least (α + ) n r edges contains a subgraph with m vertices and at least (α + c) m r edges. A result of Erdős, Stone and Simonovits implies that every α ∈ [0, 1) is a jump for r = 2. For r 3, Erdős asked whether the same is true and showed that every α ∈ [0, r! r r ) is a jump. Frankl and Rödl gave a negative answer by showing that 1 − 1 l r−1 is not a jump for r if r 3 and l > 2r. Another well-known question of Erdős is whether r! r r is a jump for r 3 and what is the smallest non-jumping number. In this paper we prove that 5 2 r! r r is not a jump for r 3. We also describe an infinite sequence of non-jumping numbers for r = 3.
Coloring Block Designs is NP-Complete. [SIAM Journal on Algebraic and Discrete Methods 3, 305 (19... more Coloring Block Designs is NP-Complete. [SIAM Journal on Algebraic and Discrete Methods 3, 305 (1982)]. Charles J. Colbourn, Marlene J. Colbourn, Kevin T. Phelps, Vojtech Rödl. Abstract. Coloring partial Steiner triple systems is shown to be NP-complete. ...
Let H1;H2;:::;Hk+1 be a sequence of k + 1 nite, undirected, simple graphs. The (mul- ticolored) R... more Let H1;H2;:::;Hk+1 be a sequence of k + 1 nite, undirected, simple graphs. The (mul- ticolored) Ramsey number r(H1;H2;:::;Hk+1) is the minimum integer r such that in every edge-coloring of the complete graph on r vertices by k + 1 colors, there is a monochromatic copy of Hi in color i for some 1 i k + 1. We describe a general technique that supplies tight lower bounds for several numbers r(H1;H2;:::;Hk+1) when k 2, and the last graph Hk+1 is the complete graph Km on m vertices. This technique enables us to determine the asymptotic behaviour of these numbers, up to a polylogarithmic factor, in various cases. In particular we show that r(K3;K3;Km) = ( m3poly logm), thus solving (in a strong form) a conjecture of Erd} os and S os raised in 1979. Another special case of our result implies that r(C4;C4;Km) = ( m2poly logm) and that r(C4;C4;C4;Km) = ( m2= log 2 m). The proofs combine combinatorial and probabilistic arguments with spectral techniques and certain esti- mates of character sums.
Szemerédi’s regularity lemma asserts that every graph can be decomposed into relatively few rando... more Szemerédi’s regularity lemma asserts that every graph can be decomposed into relatively few random-like subgraphs. This random-like behavior enables one to find and enumerate subgraphs of a given isomorphism type, yielding the so-called counting lemma for graphs. The combined application of these two lemmas is known as the regularity method for graphs and has proved useful in graph theory, combinatorial
Bulletin of the American Mathematical Society, 1977
A generalization of the Ramsey theorem is stated. This solves a problem of P. Erdös and others. T... more A generalization of the Ramsey theorem is stated. This solves a problem of P. Erdös and others. The result has recent applications in the theory of ultrafilters and model theory.
A super (d; )-regular graph on 2n vertices is a bipartite graph on the classes of vertices V1 and... more A super (d; )-regular graph on 2n vertices is a bipartite graph on the classes of vertices V1 and V2, wherejV1j =jV2j = n, in which the minimum degree and the maximum degree are between (d )n and (d + )n, and for every U V1;W V2 withjUj
Let $\R$ be the set of all finite graphs $G$ with the Ramsey property that every coloring of the ... more Let $\R$ be the set of all finite graphs $G$ with the Ramsey property that every coloring of the edges of $G$ by two colors yields a monochromatic triangle. In this paper we establish a sharp threshold for random graphs with this property. Let $G(n,p)$ be the random graph on $n$ vertices with edge probability $p$. We prove that there
A short proof of the following result of Brown and Buhler is given: For any ϵ > 0 there exists... more A short proof of the following result of Brown and Buhler is given: For any ϵ > 0 there exists n0 = n0(ϵ) such that if A is an abelian group of odd order |A| > n0 and B ⊆ A with |B| > ϵ|A|, then B must contain three distinct elements x, y, z satisfying x + y = 2z.
There are many “Ramsey theorems”, the most famous of which are Schur’s theorem (see Über die Kong... more There are many “Ramsey theorems”, the most famous of which are Schur’s theorem (see Über die Kongruenz x m +y m ≡z m (modp), Jahresber. Dtsch. Math.-Ver. 25, 114-116 (1916)), van der Waerden’s theorem (see B. L. van der Waerden, Beweis einer Baudetschen Vermutung, Nieuw Arch. Wisk. 15, 212-216 (1927)), and Ramsey’s theorem (see F. P. Ramsey, On a problem in formal logic, Proc. London Math. Soc. (2) 30, 264-285 (1930)). These theorems all all deal with partitions of the infinite set of natural numbers. In the present paper the authors are concerned with theorems of a similar kind, except that they now deal with partitions of a finite set of natural numbers {1,2,...,N}.
Let H1, H2, . . ., H k+1 be a sequence of k + 1 finite, undirected, simple graphs. The (multicolo... more Let H1, H2, . . ., H k+1 be a sequence of k + 1 finite, undirected, simple graphs. The (multicolored) Ramsey number r(H1, H2, . . . , H k+1 ) is the minimum integer r such that in every edge-coloring of the complete graph on r vertices by k+1 colors, there is a monochromatic copy of Hi in color i for some 1 ≤ i ≤ k + 1. We describe a general technique that supplies tight lower bounds for several numbers r(H1, H2, . . . , H k+1 ) when k ≥ 2, and the last graph H k+1 is the complete graph Km on m vertices. This technique enables us to determine the asymptotic behaviour of these numbers, up to a polylogarithmic factor, in various cases. In particular we show that r(K3, K3, Km) = Θ(m 3 poly log m), thus solving (in a strong form) a conjecture of Erdős and Sós raised in 1979. Another special case of our result implies that r(C4, C4, Km) = Θ(m 2 poly log m) and that r(C4, C4, C4, Km) = Θ(m 2 / log 2 m). The proofs combine combinatorial and probabilistic arguments with spectral techniques and certain estimates of character sums.
We develop a complexity theory based on the concept of the graph instead of the Boolean function.... more We develop a complexity theory based on the concept of the graph instead of the Boolean function. We show its relation to the Boolean complexity and prove some lower bounds to the complexity of explicitly given graphs.
A random construction gives new examples of simple hypergraphs with high chromatic number that ha... more A random construction gives new examples of simple hypergraphs with high chromatic number that have few edges and/or low maximum degree. In particular, for every integers k ≥ 2, r ≥ 2, and g ≥ 3, there exist r-uniform non-k-colorable hypergraphs of girth at least g with maximum degree at most r k r−1 ln k . This is only 4r 2 ln k times greater than the lower bound by Erdős and Lovász (Colloquia Math Soc János Bolyai 10 (1973), 609-627) that holds for all hypergraphs (without restrictions on girth).
We show that for any k, m, p, c, if G is a K k -free graph on N then there is an independent set ... more We show that for any k, m, p, c, if G is a K k -free graph on N then there is an independent set of vertices in G that contains an (m, p, c)-set. Hence if G is a K k -free graph on N, then one can solve any partition regular system of equations in an independent set. This is a common generalization of partition regularity theorems of Rado (who characterized systems of linear equations Ax = 0 a solution of which can be found monochromatic under any finite coloring of N) and Deuber (who provided another characterization in terms of (m, p, c)-sets and a partition theorem for them), and of Ramsey's theorem itself.
For a fixed 3-uniform hypergraph F, call a hypergraph F-free if it contains no subhypergraph isom... more For a fixed 3-uniform hypergraph F, call a hypergraph F-free if it contains no subhypergraph isomorphic to F. Let ex(n, F) denote the size of a largest F-free hypergraph G ⊆ [n] 3 . Let F n (F) denote the number of distinct labelled F-free G ⊆ [n] 3 . We show that F n (F) = 2 ex(n,F )+o(n 3 ) , and discuss related problems.
Let r 2 be an integer. The real number α ∈ [0, 1] is a jump for r if there exists c > 0 such that... more Let r 2 be an integer. The real number α ∈ [0, 1] is a jump for r if there exists c > 0 such that for every positive and every integer m r, every r-uniform graph with n > n 0 ( , m) vertices and at least (α + ) n r edges contains a subgraph with m vertices and at least (α + c) m r edges. A result of Erdős, Stone and Simonovits implies that every α ∈ [0, 1) is a jump for r = 2. For r 3, Erdős asked whether the same is true and showed that every α ∈ [0, r! r r ) is a jump. Frankl and Rödl gave a negative answer by showing that 1 − 1 l r−1 is not a jump for r if r 3 and l > 2r. Another well-known question of Erdős is whether r! r r is a jump for r 3 and what is the smallest non-jumping number. In this paper we prove that 5 2 r! r r is not a jump for r 3. We also describe an infinite sequence of non-jumping numbers for r = 3.
Coloring Block Designs is NP-Complete. [SIAM Journal on Algebraic and Discrete Methods 3, 305 (19... more Coloring Block Designs is NP-Complete. [SIAM Journal on Algebraic and Discrete Methods 3, 305 (1982)]. Charles J. Colbourn, Marlene J. Colbourn, Kevin T. Phelps, Vojtech Rödl. Abstract. Coloring partial Steiner triple systems is shown to be NP-complete. ...
Let H1;H2;:::;Hk+1 be a sequence of k + 1 nite, undirected, simple graphs. The (mul- ticolored) R... more Let H1;H2;:::;Hk+1 be a sequence of k + 1 nite, undirected, simple graphs. The (mul- ticolored) Ramsey number r(H1;H2;:::;Hk+1) is the minimum integer r such that in every edge-coloring of the complete graph on r vertices by k + 1 colors, there is a monochromatic copy of Hi in color i for some 1 i k + 1. We describe a general technique that supplies tight lower bounds for several numbers r(H1;H2;:::;Hk+1) when k 2, and the last graph Hk+1 is the complete graph Km on m vertices. This technique enables us to determine the asymptotic behaviour of these numbers, up to a polylogarithmic factor, in various cases. In particular we show that r(K3;K3;Km) = ( m3poly logm), thus solving (in a strong form) a conjecture of Erd} os and S os raised in 1979. Another special case of our result implies that r(C4;C4;Km) = ( m2poly logm) and that r(C4;C4;C4;Km) = ( m2= log 2 m). The proofs combine combinatorial and probabilistic arguments with spectral techniques and certain esti- mates of character sums.
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Papers by Vojtěch Rödl