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We prove that among all flag triangulations of manifolds of odd dimension    2  r  ‐  1   , with a sufficient number of vertices, the unique maximizer of the entries of the   f  ‐,   h  ‐,   g  ‐ and   γ  ‐vector is the balanced join of   r   cycles. Our proof uses methods from extremal graph theory.

UPPER BOUND THEOREM FOR ODD‐DIMENSIONAL FLAG TRIANGULATIONS OF MANIFOLDS