Abstract. We investigate minimum vertex degree conditions for 3-uniform hypergraphs which ensure the existence of loose Hamilton cycles. A loose Hamilton cycle is a spanning cycle in which only consecutive edges intersect and these intersections consist of precisely one vertex.We prove that every 3-uniform n-vertex (n even) hypergraph H with minimum vertex degree δ 1 pHq ě`7 16`o p1q˘`n 2˘c ontains a loose Hamilton cycle. This bound is asymptotically best possible. §1. IntroductionWe consider k-uniform hypergraphs H " pV, Eq with vertex sets V " V pHq and edge sets E " EpHq Ď`V k˘, where`V k˘d enotes the family of all k-element subsets of the set V . We often identify a hypergraph H with its edge set, i.e., H Ď`V k˘, and for an edge tv 1 , . . . , v k u P H we often suppress the enclosing braces and write v 1 . . . v k P H instead. Given a k-uniform hypergraph H " pV, Eq and a set S " tv 1 , . . . , v s u P`V sl et degpSq " degpv 1 , . . . , v s q denote the number of edges of H containing the set S and let N pSq " N pv 1 , . . . , v s q denote the set of those pk´sq-element sets T P`V k´s˘s uch that S Y T forms an edge in H. We denote by δ s pHq the minimum s-degree of H, i.e., the minimum of degpSq over all s-element sets S Ď V . For s " 1 the corresponding minimum degree δ 1 pHq is referred to as minimum vertex degree whereas for s " k´1 we call the corresponding minimum degree δ k´1 pHq the minimum collective degree of H.We study sufficient minimum degree conditions which enforce the existence of spanning, so-called Hamilton cycles. A k-uniform hypergraph C is called an -cycle if there is a cyclic ordering of the vertices of C such that every edge consists of k consecutive vertices, every vertex is contained in an edge and two consecutive edges (where the ordering of the edges 2010 Mathematics Subject Classification. 05C65 (primary), 05C45 (secondary).