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This paper thoroughly explores for the first time the constraining power of the void size function on the properties of dark energy (DE) from a survey mock catalogue, the official <jats:italic>Euclid<\/jats:italic> Flagship simulation. We identified voids in the Flagship light-cone, which closely matches the features of the upcoming <jats:italic>Euclid<\/jats:italic> spectroscopic data set. We modelled the void size function considering a state-of-the art methodology: we relied on the volume-conserving (Vdn) model, a modification of the popular Sheth &amp; van de Weygaert model for void number counts, extended by means of a linear function of the large-scale galaxy bias. We found an excellent agreement between model predictions and measured mock void number counts. We computed updated forecasts for the <jats:italic>Euclid<\/jats:italic> mission on DE from the void size function and provided reliable void number estimates to serve as a basis for further forecasts of cosmological applications using voids. We analysed two different cosmological models for DE: the first described by a constant DE equation of state parameter, <jats:italic>w<\/jats:italic>, and the second by a dynamic equation of state with coefficients <jats:italic>w<\/jats:italic><jats:sub>0<\/jats:sub> and <jats:italic>w<\/jats:italic><jats:sub><jats:italic>a<\/jats:italic><\/jats:sub>. We forecast 1<jats:italic>\u03c3<\/jats:italic> errors on <jats:italic>w<\/jats:italic> lower than 10% and we estimated an expected figure of merit (FoM) for the dynamical DE scenario FoM<jats:sub><jats:italic>w<\/jats:italic><jats:sub>0<\/jats:sub>,\u2006<jats:italic>w<\/jats:italic><jats:sub><jats:italic>a<\/jats:italic><\/jats:sub><\/jats:sub>\u2004=\u200417 when considering only the neutrino mass as additional free parameter of the model. The analysis is based on conservative assumptions to ensure full robustness, and is a pathfinder for future enhancements of the technique. Our results showcase the impressive constraining power of the void size function from the <jats:italic>Euclid<\/jats:italic> spectroscopic sample, both as a stand-alone probe, and to be combined with other <jats:italic>Euclid<\/jats:italic> cosmological probes.<\/jats:p>","DOI":"10.1051\/0004-6361\/202244095","type":"journal-article","created":{"date-parts":[[2022,8,19]],"date-time":"2022-08-19T18:47:36Z","timestamp":1660934856000},"page":"A162","source":"Crossref","is-referenced-by-count":56,"title":["<i>Euclid<\/i>: Cosmological forecasts from the void size function"],"prefix":"10.1051","volume":"667","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9843-723X","authenticated-orcid":false,"given":"S.","family":"Contarini","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1886-8348","authenticated-orcid":false,"given":"G.","family":"Verza","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6146-4437","authenticated-orcid":false,"given":"A.","family":"Pisani","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0876-2101","authenticated-orcid":false,"given":"N.","family":"Hamaus","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0973-4804","authenticated-orcid":false,"given":"M.","family":"Sahl\u00e9n","sequence":"additional","affiliation":[]},{"given":"C.","family":"Carbone","sequence":"additional","affiliation":[]},{"given":"S.","family":"Dusini","sequence":"additional","affiliation":[]},{"given":"F.","family":"Marulli","sequence":"additional","affiliation":[]},{"given":"L.","family":"Moscardini","sequence":"additional","affiliation":[]},{"given":"A.","family":"Renzi","sequence":"additional","affiliation":[]},{"given":"C.","family":"Sirignano","sequence":"additional","affiliation":[]},{"given":"L.","family":"Stanco","sequence":"additional","affiliation":[]},{"given":"M.","family":"Aubert","sequence":"additional","affiliation":[]},{"given":"M.","family":"Bonici","sequence":"additional","affiliation":[]},{"given":"G.","family":"Castignani","sequence":"additional","affiliation":[]},{"given":"H. 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