## 1 Introduction  
In the last few years, the use of components produced through additive manufacturing (AM) processes is rapidly growing in many industrial applications. AM parts are currently employed in applications subjected to static loads at room and high temperature [1, 2], such as brackets [3], medical devices [4], or heat exchangers [5]; however, their use is still limited for applications subjected to fatigue loads [6, 7]. Among the employed alloys, the ones belonging to the Al-Sisystem are drawing considerable interest thanks to their good mechanical properties and low density. The resulting microstructure of Al-Si produced by AM is much finer than the microstructure obtained by conventional casting, thanks to the high cooling rates associated to the solidification of a limited liquid pool during the layer-by-layer manufacturing. Indeed, cast alloys display a coarse microstructure with eutectic Si appearing as brittle large plates or needles [8]. On the other hand, the AMed microstructure of Al-Si alloys obtained by selective laser melting (SLM) or laser metal wire deposition [9] consists of an interconnected Al-Si eutectic structure, with small grains distributed inside superimposed melt pools. These features, together with the high powder purity that ensures the absence of the detrimental Fe-rich inclusions/precipitates, has been reported to positively affect the material mechanical resistance, in both static and cyclic conditions. However, the high surface roughness and the large defects originating during the building process are detrimental for the fatigue response of as-built AM parts and prevent their use in many structural applications. University and industry research efforts (see [6; 7] and the references therein) are therefore currently focusing on the mechanisms of damage and failure of AM parts subjected to fatigue loads and on the effect of process parameters and post-process treatments on the fatigue behavior, with the aim of proposing reliable design methodologies [10] that would guarantee the structural integrity of AM parts and would satisfy the safety standard required in many critical structural applications. Many research and experimental tests have been carried out in the literature to investigate the fatigue response of AM parts, focusing mainly on the low cycle fatigue (LCF) [11] and the high cycle fatigue (HCF) response [12; 13]. Regarding the HCF response, the effect of process parameters [14], heat treatments [15], defect size, and morphology [16] has been widely investigated. Moreover, there is increasing interest on the very high cycle fatigue (VHCF) behavior of AM parts, since in many structural applications the required fatigue life is significantly larger than that assessed through HCF tests. In particular, the VHCF response of AM Ti6 Al4V [17; 18], AlSi12 [19], and AlSi10 Mg [20; 21; 22] alloys has been recently investigated in the literature. As for the AM AlSi10 Mg alloy, the VHCF response up to \(10^{9}\) cycles of as-built and stress-relieved Gaussian specimens with horizontal building direction was studied in [20; 21], respectively. [PERSON] et al. [22] investigated the VHCF response up to \(10^{8}\) cycles of AM AlSi10 Mg hourglass specimens with a vertical building direction. In particular, in [22], the positive effect of platform heating, which affects the defect morphology and microstructure, was experimentally assessed, and a pore-crack interaction mechanism during crack propagation was shown through X-ray computed tomography. At present, even though it is well-known that size effects significantly influence the VHCF response of wrought [23] and AM parts [24], no experimental tests have been carried out on AM AlSi10 Mg specimens with a large risk volume (i.e., the volume of the material subjected to a stress amplitude larger than 90% of the maximum applied stress [25]) and a vertical building direction. In this paper, the VHCF response of selective laser melting (SLM) AlSi10 Mg specimens with a built in vertical direction is experimentally assessed.