This paper aims to compare the safe-haven roles of gold and Bitcoin for energy commodities, including oils and petroleum, during COVID-19. Specifically, we examine the presence of reduction in downside risk after mixing gold/Bitcoin with such energy commodities. To do this, we account for dependence among energy commodities and gold/Bitcoin returns by applying a (vine) copula. The findings show that gold substantially reduces the downside risk of a portfolio containing any allocation to gold and energy commodities, indicating its safe-haven ability. In contrast, Bitcoin’s safe-haven functionality is inconsistent since the downside risk reduction is achieved for Bitcoin’s small allocation only.

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We study the non-rigidity of Euclidean t-designs, namely we study when Euclidean designs (in particular certain tight Euclidean designs) can be deformed keeping the property of being Euclidean t-designs. We show that certain tight Euclidean t-designs are non-rigid, and in fact satisfy a stronger form of non-rigidity which we call strong nonrigidity. This shows that there are plenty of non-isomorphic tight Euclidean t-designs for certain parameters, which seems to have been unnoticed before. We also include the complete classification of tight Euclidean 2-designs.

Let F, G, and H be non-empty graphs. The notation F → (G, H) means that if all edges of F are arbitrarily colored by red or blue, then either the subgraph of F induced by all red edges contains a graph G or the subgraph of F induced by all blue edges contains a graph H. A graph F satisfying two conditions: F → (G, H) and for every e ∈ E(F), (F − e) (G, H) is called a Ramsey (G, H)−minimal graph. In this paper, we determine all non-isomorphic Ramsey (2K 2 , K 4)−minimal graphs.