Fan-goodness of sparse graphs

Let G be a connected graph of order n, Fₖ be a fan consisting of k triangles sharing a common vertex, and tFₖ be t vertex-disjoint copies of Fₖ. Brennan (2017) showed the Ramsey number r(G,Fₖ)=2n-1 for G being a unicyclic graph for n ≥ k²-k+1 and k≥ 18, and asked the threshold $c(n)$ for which r(G,Fₖ) ≥ 2n holds for any G containing at least $c(n)$ cycles and n being large. In this paper, we consider fan-goodness of general sparse graphs and show that if G has at most $n(1+ε(k))$ edges, where $ε(k)$ is a constant depending on k, then r(G,Fₖ)=2n-1 for n≥ 36k⁴, which implies that $c(n)$ is greater than $ε(k) n$. Moreover, if G has at most $n(1+ε(k,t))$ edges, where $ε(k,t)$ is a constant depending on $k,t$, then r(G,tFₖ)=2n+t-2 provided n≥ 161t²k⁴.