arXiv subjects

Diana Piguet

Publications and source records attributed to Diana Piguet.

24 records · Page 2Linked to original sources

arXiv2012

2-Colored Matchings in a 3-Colored K^{3}_{12}

Let $K_{n}^{r}$ denote the complete $r$-uniform hypergraph on $n$ vertices. A matching $M$ in a hypergraph is a set of pairwise vertex disjoint edges. Recent Ramsey-type results rely on lemmas about the size of monochromatic matchings. A starting point for this study comes from a well-known result of Alon, Frankl, and Lovász (1986). Our motivation is to find the smallest $n$ such that every $t$-coloring of $K_{n}^{r}$ contains an $s$-colored matching of size $k$. It has been conjectured that in every coloring of the edges of $K_n^r$ with 3 colors there is a 2-colored matching of size at least $k$ provided that $n \geq kr + \lfloor \frac{k-1}{r+1} \rfloor$. The smallest test case is when $r=3$ and $k=4$. We prove that in every 3-coloring of the edges of $K_{12}^3$ there is a 2-colored matching of size 4.

math.CO↗

arXiv2011

Turánnical hypergraphs

This paper is motivated by the question of how global and dense restriction sets in results from extremal combinatorics can be replaced by less global and sparser ones. The result we consider here as an example is Turan's theorem, which deals with graphs G=([n],E) such that no member of the restriction set consisting of all r-tuples on [n] induces a copy of K_r. Firstly, we examine what happens when this restriction set is replaced just by all r-tuples touching a given m-element set. That is, we determine the maximal number of edges in an n-vertex such that no K_r hits a given vertex set. Secondly, we consider sparse random restriction sets. An r-uniform hypergraph R on vertex set [n] is called Turannical (respectively epsilon-Turannical), if for any graph G on [n] with more edges than the Turan number ex(n,K_r) (respectively (1+\eps)ex(n,K_r), no hyperedge of R induces a copy of K_r in G. We determine the thresholds for random r-uniform hypergraphs to be Turannical and to epsilon-Turannical. Thirdly, we transfer this result to sparse random graphs, using techniques recently developed by Schacht [Extremal results for random discrete structures] to prove the Kohayakawa-Luczak-Rodl Conjecture on Turan's theorem in random graphs.

math.CO↗

arXiv2011

An approximate version of the Loebl-Komlos-Sos conjecture

Loebl, Komlos, and Sos conjectured that if at least half of the vertices of a graph G have degree at least some natural number k, then every tree with at most k edges is a subgraph of G. Our main result is an approximate version of this conjecture for large enough n=|V(G)|, assumed that n=O(k). Our result implies an asymptotic bound for the Ramsey number of trees. We prove that r(T_k,T_m)\leq k+m+o(k+m),as k+m tends to infinity.

math.CO↗

arXiv2010

Finite trees are Ramsey under topological embeddings

We show that the class of finite rooted binary plane trees is a Ramsey class (with respect to topological embeddings that map leaves to leaves). That is, for all such trees P,H and every natural number k there exists a tree T such that for every k-coloring of the (topological) copies of P in T there exists a (topological) copy H' of H in T such that all copies of P in H' have the same color. When the trees are represented by the so-called rooted triple relation, the result gives rise to a Ramsey class of relational structures with respect to induced substructures.

math.CO↗

arXiv2009

The tripartite Ramsey number for trees

We prove that for all epsilon>0 there are c>0 and n_0 such that for all n>n_0 the following holds. For any two-colouring of the edges of $K_{n,n,n}$ one colour contains copies of all trees T of order t<(3-epsilon)n/2 and with maximum degree at most n^c. This confirms a conjecture of Schelp.

math.CO↗

arXiv2007

The Loebl-Komlos-Sos conjecture for trees of diameter 5 and for certain caterpillars

Loebl, Komlos, and Sos conjectured that if at least half the vertices of a graph G have degree at least some k, then every tree with at most k edges is a subgraph of G. We prove the conjecture for all trees of diameter at most 5 and for a class of caterpillars. Our result implies a bound on the Ramsey number r(T,F) of trees T, F from the above classes.

math.CO↗