The projective dimension of three cubics is at most 5

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Let R be a polynomial ring over a field and I an ideal generated by three forms of degree three. Motivated by Stillman's question, Engheta proved that the projective dimension pd(R/I) of R/I is at most 36, although the example with largest projective dimension he constructed has pd(R/I)=5. Based on computational evidence, it had been conjectured that pd(R/I)≤ 5. In the present paper we prove this conjectured sharp bound.