Upper bound for r(F_s, F_t) when 5 ≤ s ≤ t

Establish an improved upper bound for the generalized Ramsey number r(F_s, F_t) for integers s and t satisfying 5 ≤ s ≤ t, where F_k denotes the graph on k vertices with vertex set {v_1, ..., v_k} and edges between v_i and v_j for all i in {1, ..., floor(k/2)} and j in {i+1, ..., k+1−i}.

Background

The paper studies generalized Ramsey numbers r(G1, G2) and focuses on the special family F_k, defined on k vertices with a structured adjacency pattern. Exact values are established for several small cases: r(F_2, F_t)=t (Proposition 2.1), r(F_3, F_t) equals t+1 if t is even and t if t is odd (Corollary 2.1), and r(F_4, F_t)=2t−1 for t≥3 (Theorem 2.4). A general lower bound r(F_s, F_t) ≥ 1+(s−1) floor(t/2) is provided (Theorem 2.2).

After summarizing known values in Table 1, the authors explicitly state that upper bounds for r(F_s, F_t) remain unknown for the range 5 ≤ s ≤ t and pose an open problem to address this gap.