Monochromatic loose paths in multicolored kk-uniform cliques

For integers $k\ge 2$ and $\ell\ge 0$, a $k$-uniform hypergraph is called a loose path of length $\ell$, and denoted by $P_\ell^{(k)}$, if it consists of $\ell$ edges $e_1,\dots,e_\ell$ such that $|e_i\cap e_j|=1$ if $|i-j|=1$ and $e_i\cap e_j=\emptyset$ if $|i-j|\ge2$. In other words, each pair of consecutive edges intersects on a single vertex, while all other pairs are disjoint. Let $R(P_\ell^{(k)};r)$ be the minimum integer $n$ such that every $r$-edge-coloring of the complete $k$-uniform hypergraph $K_n^{(k)}$ yields a monochromatic copy of $P_\ell^{(k)}$. In this paper we are mostly interested in constructive upper bounds on $R(P_\ell^{(k)};r)$, meaning that on the cost of possibly enlarging the order of the complete hypergraph, we would like to efficiently find a monochromatic copy of $P_\ell^{(k)}$ in every coloring. In particular, we show that there is a constant $c>0$ such that for all $k\ge 2$, $\ell\ge3$, $2\le r\le k-1$, and $n\ge k(\ell+1)r(1+\ln(r))$, there is an algorithm such that for every $r$-edge-coloring of the edges of $K_n^{(k)}$, it finds a monochromatic copy of $P_\ell^{(k)}$ in time at most $cn^k$. We also prove a non-constructive upper bound $R(P_\ell^{(k)};r)\le(k-1)\ell r$

Novel placodont material and paleoenvironment analysis of Triassic deposits of Rocha da Pena (Algarve, southern Portugal)

The Triassic deposits of the Algarve region have been known for almost 200 years with the first vertebrate fossil descriptions published in 1976 by Palain and in 1977 by Russell and Russell. It was followed by the discovery of the monospecific bonebed in Rocha da Pena. The bonebed bears numerous temnospondyl remains, which in 2015 were assigned to a novel species Metoposaurus algarvensis. The occurrence of that taxon suggested a chronostratigraphic range between Carnian and middle Norian. During the following fieldwork, the first Portuguese placodont material was recovered. Based on osteoderm morphology was assigned to genus Henodus, which previously was exclusively known from lower Carnian deposits in Lustnau in southern Germany. Since then new Portuguese fossil material has been recovered including an isolated, nearly complete skull, which is described herein. Based on several features (flat and broad skull with short and broad rostrum composed of maxillae and premaxillae, rectangular shape of the skull, toothless maxillae substituted by longitudinally extending curved groove and non-contacting palatines), the specimen is assigned to the genus Henodus, confirming the occurrence of this taxon in Portugal. This extends its paleogeographic range up till western Laurasia. The occurrence of this taxon in Rocha da Pena may indicate either age of lower Carnian for the local deposits or an extended chronostratigraphic distribution of Henodus. The described specimen comes from a fossiliferous layer that contains isolated remains of placodonts (most probably Henodus), actinopterygian fishes, and the first occurrence of hybodont sharks in the Triassic deposits of Portugal. The gathered sedimentological data-enabled interpretation of the paleoenvironment of the studied sections. The studied deposits can most likely be attributed to a marginal environment, with the transition from playa to near-shore or shallow lagoon.Os depósitos triásicos do Algarve são conhecidos há quase 200 anos com as primeiras descrições de fósseis de vertebrados publicadas em 1976 por Palain e em 1977 por Russell e Russell. Seguiu-se a descoberta de uma cadada de ossos, monoespecífica, na Rocha da Pena, que apresenta inúmeros vestígios de temnospôndilos, que em 2015 foram atribuídos a uma nova espécie Metoposaurus algarvensis. A ocorrência desse táxon sugeriu uma idade entre Carnian e Norian médio. Durante as escavações seguintes, o primeiro material de placodonte português foi recuperado e com base na morfologia da osteoderme foi atribuído ao género Henodus, que anteriormente era conhecido exclusivamente de depósitos carnianos inferiores em Lustnau no sul da Alemanha. Desde então, novo material fóssil foi recuperado, incluindo um crânio quase completo isolado, que é descrito aqui. Com base em várias características (crânio plano e largo com rostro curto e largo composto de maxilares e pré-maxilares, forma retangular do crânio, maxilas desdentadas substituídas por sulco curvo que se prolonga longitudinalmente e palatinos sem contato), o espécime é classificado como género Henodus, confirmando a ocorrência deste táxon em Portugal. Isso estende seu alcance paleogeográfico até o oeste da Laurásia. A ocorrência deste táxon na Rocha da Pena pode indicar uma idade de baixo Carnian para os depósitos locais ou uma distribuição estratigráfica estendida de Henodus. O espécime descrito provém de uma camada fossilífera que contém restos isolados de placodontes (muito provavelmente Henodus), peixes actinopterígios e a primeira ocorrência de tubarões hibodontiformes nos depósitos triásicos de Portugal. Os dados sedimentológicos permitiram a interpretação do paleoambiente das seções estudadas como um ambiente marginal mais provável, com transição de playa para próximo à costa ou ambiente de lagoa rasa

On the order of the largest induced tree in a random graph

AbstractConsider a random graph K(n, p) with n labeled vertices in which the edges are chosen independently and with a probability p. Let Tn(p) be the order of the largest induced tree in K(n, p). Among other results it is shown, using an algorithmic approach, that if p=(c log n)/n, where c ≥ e is a constant, then for any fixed ε > 01c−εlog lognlognn<Tn(p)<2c+εlog lognlogn almost surely