Monochromatic cycle power partitions

Improving our earlier result we show that for every integer k≥1 there exists a c(k) such that in every 2-colored complete graph apart from at most c(k) vertices the vertex set can be covered by 200k2logk vertex disjoint monochromatic kth powers of cycles. © 2016 Elsevier B.V

Rainbow matchings in bipartite multigraphs

Suppose that $k$ is a non-negative integer and a bipartite multigraph $G$ is the union of $$N=\left\lfloor \frac{k+2}{k+1}n\right\rfloor -(k+1)$$ matchings $M_1,\dots,M_N$, each of size $n$. We show that $G$ has a rainbow matching of size $n-k$, i.e. a matching of size $n-k$ with all edges coming from different $M_i$'s. Several choices of parameters relate to known results and conjectures

An extension of the Ruzsa-Szemerédi theorem

We let G (r) (n,m) denotethesetofr-uniform hypergraphs with n vertices and m edges, and f (r) (n,p,s) is the smallest m such that every member of G (r) (n,m) containsamember of G (r) (p,s). In this paper we are interested in fixed values r, p and s for which f (r) (n,p,s) grows quadratically with n. A probabilistic construction of Brown, Erdős and T. Sós ([2]) implies that f (r) (n,s(r − 2) + 2,s)=Ω(n 2). In the other direction the most interesting question they could not settle was whether f (3) (n,6,3) = o(n 2). This was proved by Ruzsa and Szemerédi [11]. Then Erdős,FranklandRödl [6] extended this result to any r: f (r) (n,3(r − 2) + 3,3) = o(n 2), and they conjectured ([4], [6]) that the Brown, Erdős and T. Sós bound is best possible in the sense that f (r) (n,s(r −2)+ 3,s)=o(n 2). In this paper by giving an extension of the Erdős, Frankl, Rödl Theorem (and thus the Ruzsa–Szemerédi Theorem) we show that indeed the Brown, Erdős, T. Sós Theorem is not far from being best possible. Our main result is f (r) (n, s(r − 2) + 2 + ⌊log 2 s⌋,s)=o(n 2). 1

Large monochromatic components in edge colored graphs with a minimum degree condition

It is well-known that in every k-coloring of the edges of the complete graph Kn there is a monochromatic connected component of order at least (formula presented)k-1. In this paper we study an extension of this problem by replacing complete graphs by graphs of large minimum degree. For k = 2 the authors proved that δ(G) ≥(formula presented) ensures a monochromatic connected component with at least δ(G) + 1 vertices in every 2-coloring of the edges of a graph G with n vertices. This result is sharp, thus for k = 2 we really need a complete graph to guarantee that one of the colors has a monochromatic connected spanning subgraph. Our main result here is that for larger values of k the situation is different, graphs of minimum degree (1 − ϵk)n can replace complete graphs and still there is a monochromatic connected component of order at least (formula presented), in fact (formula presented) suffices. Our second result is an improvement of this bound for k = 3. If the edges of G with δ(G) ≥ (formula presented) are 3-colored, then there is a monochromatic component of order at least n/2. We conjecture that this can be improved to 9 and for general k we (onjectu) the following: if k ≥ 3 and G is a graph of order n such that δ(G) ≥ (formula presented) n, then in any k-coloring of the edges of G there is a monochromatic connected component of order at least (formula presented). © 2017, Australian National University. All rights reserved

Partitioning 3-colored complete graphs into three monochromatic cycles

We show in this paper that in every 3-coloring of the edges of Kn all but o(n) of its vertices can be partitioned into three monochromatic cycles. From this, using our earlier results, actually it follows that we can partition all the vertices into at most 17 monochromatic cycles, improving the best known bounds. If the colors of the three monochromatic cycles must be different then one can cover ( 3 4 − o(1))n vertices and this is close to best possible

Long rainbow cycles in proper edge-colorings of complete graphs

We show that any properly edge-colored Kn contains a rainbow cycle with at least (4=7 − o(1))n edges. This improves the lower bound of n=2 − 1 proved in [1]

Turan and Ramsey numbers in linear triple systems II

In this paper we continue our studies of Turan and Ramsey numbers in linear triple systems, defined as 3-uniform hypergraphs in which any two triples intersect in at most one vertex. In [7] the two main problems left open were the Turan number of the wicket and the Ramsey property of the sail. In this paper we present some progress towards both of these problems.(c) 2022 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/)

On the multi-colored ramsey numbers of paths and even cycles

In this paper we improve the upper bound on the multi-color Ramsey numbers of paths and even cycles. More precisely, we prove the following. For every r ≥ 2 there exists an n0 = n0(r) such that for n ≥ n0 we have (Formula Presented). For every r ≥ 2 and even n we have (Formula Presented). The main tool is a stability version of the Erdős-Gallai theorem that may be of independent interest. © 2016 Australian National University. All rights reserved