The Ramsey number of loose paths in 3-uniform hypergraphs

Recently, asymptotic values of 2-color Ramsey numbers for loose cycles and also loose paths were determined. Here we determine the 2-color Ramsey number of 3-uniform loose paths when one of the paths is significantly larger than the other: for every $n\geq \Big\lfloor\frac{5m}{4}\Big\rfloor$, we show that $$R(\mathcal{P}^3_n,\mathcal{P}^3_m)=2n+\Big\lfloor\frac{m+1}{2}\Big\rfloor.$

Counting triangles in some Ramsey graphs

We extend Goodman’s result on the cardinality of monochromatic triangles in a 2-colored complete graph to the case of bounding the number of triangles in the first color. We apply it to derive the upper bounds on some non-diagonal Ramsey numbers. In particular we show that R(K4 − e,K8) \u3c= 45