Vojtĕch Rödl

The Ramsey number for hypergraph cycles I

Let Cn denote the 3-uniform hypergraph loose cycle, that is the hypergraph with vertices v1.....,vn and edges v1v2v3, v3v4v5, v5v6v7,.....,vn-1vnv1. We prove that every red-blue colouring of the edges of the complete 3-uniform hypergraph with N vertices contains a monochromatic copy of Cn, where N is asymptotically equal to 5n/4. Moreover this result is (asymptotically) best possible.

Sharp Bounds For Some Multicolor Ramsey Numbers

Let H1,H2, . . .,Hk+1 be a sequence of k+1 finite, undirected, simple graphs. The (multicolored) 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) = Θ(m3 poly logm), thus solving (in a strong form) a conjecture of Erdőos and Sós raised in 1979. Another special case of our result implies that r(C4,C4,Km) = Θ(m2 poly logm) and that r(C4,C4,C4,Km) = Θ(m2/log2m). The proofs combine combinatorial and probabilistic arguments with spectral techniques and certain estimates of character sums.

The ramsey number for 3-uniform tight hypergraph cycles

Let C(3)n denote the 3-uniform tight cycle, that is, the hypergraph with vertices v1,.–.–., vn and edges v1v2v3, v2v3v4,.–.–., vn−1vnv1, vnv1v2. We prove that the smallest integer N = N(n) for which every red–blue colouring of the edges of the complete 3-uniform hypergraph with N vertices contains a monochromatic copy of C(3)n is asymptotically equal to 4n/3 if n is divisible by 3, and 2n otherwise. The proof uses the regularity lemma for hypergraphs of Frankl and Rodl.

Monochromatic Vs multicolored paths

Letl andk be positive integers, and letX={0,1,...,lk−1}. Is it true that for every coloring δ:X×X→{0,1,...} there either exist elementsx0<x1<...<xl ofX with δ(x0,x1)=δ(x1,x2)=...=δ(xl−1,xl), or else there exist elementsy0<y1<...<yk ofX with δ(yi−1,yi) ∈ δ(yj−1,yj) for all 1<-i<j≤k? We prove here that this is the case if eitherl≤2, ork≤4, orl≥(3k)2k. The general question remains open.

On ramsey families of sets

The main result of this paper is a lemma which can be used to prove the existence of highchromatic subhypergraphs of large girth in various hypergraphs. In the last part of the paper we use amalgamation techniques to prove the existence for everyl, k ≥ 3 of a setA of integers such that the hypergraph having as edges all the arithmetic progressions of lengthk inA has both chromatic number and girth greater thanl.

Distance preserving ramsey graphs

We prove the following metric Ramsey theorem. For any connected graph G endowed with a linear order on its vertex set, there exists a graph R such that in every colouring of the t-sets of vertices of R it is possible to find a copy G* of G inside R satisfying: distG*(x, y) = distR(x, y) for every x, y ∈ V(G*); the colour of each t-set in G* depends only on the graph-distance metric induced in G by the ordered t-set.

Counting subgraphs in quasi-random 4-uniform hypergraphs: Counting Subgraphs In Quasi-Random 4-Uniform Hypergraphs

A bipartite graph G = (V1 ∪ V2, E) is (δ, d)-regular if ̨̨ d− d(V ′ 1 , V ′ 2 ) ̨̨ < δ whenever V ′ i ⊂ Vi, |V ′ i | ≥ δ|Vi|, i = 1, 2. Here, d(V ′ 1 , V ′ 2 ) = e(V ′ 1 , V ′ 2 )/|V ′ 1 ||V ′ 2 | stands for the density of the pair (V ′ 1 , V ′ 2 ). An easy counting argument shows that if G = (V1 ∪ V2 ∪ V3, E) is a 3-partite graph whose restrictions on V1 ∪V2, V1 ∪V3, V2 ∪V3 are (δ, d)regular, then G contains (d ± f(δ))|V1||V2||V3| copies of K3. This fact and its various extensions are the key ingredients in most applications of Szemerédi’s Regularity Lemma. To derive a similar results for r-uniform hypergraphs, r > 2, is a harder problem. In 1994, Frankl and Rödl developed a regularity lemma and counting argument for 3-uniform hypergraphs. In this paper, we exploit their approach to develop a counting argument for 4-uniform hypergraphs.

On Systems of Small Sets with No Large Δ-Subsystems

A family of k sets is called a Δ-system if any two sets have the same intersection. Denote by f(r, k) the least integer so that any r-uniform family of f(r, k) sets contains a Δ-system consisting of k sets. We prove that, for every fixed r, f(r, k) = kr + o(kr). Using a recent result of Molloy and Reed [5], a bound on the error term is provided for sufficiently large k.

Note on the point character of l 1-spaces

We prove that for any ordinal α, any integer t ≥ 0, the point character of the space l1(ωα + t) is no more than ωα. Combined with an earlier result from [5], this yields that for any infinite cardinal κ the point character of l1(κ) is the largest cardinal ωα ≤ κ where α = 0 or a limit ordinal.