Finite dimensional semigroup quadratic algebras with minimal number of relations

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A quadratic semigroup algebra is an algebra over a field given by the generators x₁,...,xₙ and a finite set of quadratic relations each of which either has the shape xⱼxₖ=0 or the shape xⱼxₖ=xₗxₘ. We prove that a quadratic semigroup algebra given by n generators and d≤ n²+n/4 relations is always infinite dimensional. This strengthens the Golod--Shafarevich estimate for the above class of algebras. Our main result however is that for every n, there is a finite dimensional quadratic semigroup algebra with n generators and δₙ relations, where δₙ is the first integer greater than n²+n/4. This shows that the above Golod-Shafarevich type estimate for semigroup algebras is sharp.