Joanna Polcyn

Short paths in quasi-random triple systems with sparse underlying graphs

The regularity lemma for 3-uniform hypergraphs asserts that every large hypergraph can be decomposed into a bounded number of quasi-random structures consisting of a sub-hypergraph and a sparse underlying graph. In this paper we show that in such a quasi-random structure most pairs of the edges of the graph can be connected by hyperpaths of length at most twelve. Some applications are also given.

One more Tur\'an number and Ramsey number for the loose 3-uniform path of length three

Abstract Let P denote a 3-uniform hypergraph consisting of 7 vertices a, b, c, d, e, f, g and 3 edges {a, b, c}, {c, d, e}, and {e, f, g}. It is known that the r-color Ramsey number for P is R(P; r) = r + 6 for r ≤ 9. The proof of this result relies on a careful analysis of the Turán numbers for P. In this paper, we refine this analysis further and compute the fifth order Turán number for P, for all n. Using this number for n = 16, we confirm the formula R(P; 10) = 16.

The multipartite Ramsey number for the 3-path of length three

We study the Ramsey number for the 3-path of length three and

Short paths in ε-regular pairs and small diameter decompositions of dense graphs

n

Short paths in 3-uniform quasi-random hypergraphs

colors and show that

On multicolor Ramsey numbers for loose k-paths of length three

R(P^3_3;n)\le \lambda_0 n+7\sqrt{n}

Subhypergraph counts in extremal and random hypergraphs and the fractional q-independence

, for some explicit constant

A tale of stars and cliques

\lambda_0=1.97466\dots

On the Multicolor Ramsey Number for 3-Paths of Length Three

.

Multicolor Ramsey Numbers and Restricted Turán Numbers for the Loose 3-Uniform Path of Length Three

In this paper we give optimal vertex degree conditions that guarantee connection by short paths in @e-regular bipartite graphs. We also study a related question of decomposing an arbitrary graph into subgraphs of small diameter.