  Pliable Index CODing (PICOD) is a variant of the Index Coding (IC) problem in
which a user is satisfied whenever it can successfully decode any one message
that is not in its side information set, as opposed to a fixed pre-determined
message. The complete--$S$ PICOD with $m$ messages, for $S\subseteq[0:m-1]$,
has $n = \sum_{s\in S} \binom{m}{s}$ users with distinct side information sets.
Past work on PICOD provided tight converse results when either the sender is
constrained to use linear codes, or for some special classes of complete--$S$
PICOD. This paper provides a tight information theoretic converse result (i.e.,
no restriction to linear codes) for the so-called "consecutive complete--$S$"
PICOD, where the set $S$ satisfies $S=[s_{min} : s_{max}]$ for some $0\leq
s_{min} \leq s_{max} \leq m-1$. This result extends existing converse results
and shows that linear codes have the smallest possible code length given by
$\min(m-s_{\min},1+s_{\max})$. The central contribution is a novel proof
technique rooted in combinatorics. The main idea is to consider all the
messages a user can eventually successfully decode, in addition to its own
desired message. This allows us to circumvent the necessity of essentially
considering all possible assignments of desired messages for the users. The
keystone of the proof is to show that, for the case of $S=\{s\}$ and $m =
2s+1$, there exists at least one user who can decode $s+1$ messages. From this,
the extension to the "consecutive complete--$S$" PICOD follows.
