Erdős's Alice–Bob clique game on $K_n$: does Bob have a winning strategy for all $n\geq 3$? (Erdős #778)

Alice and Bob alternately colour the edges of the complete graph $K_n$, Alice colouring red and Bob colouring blue, Alice moving first, until every edge is coloured. Three games are posed. (1) Alice wins if, at the end, her largest red clique (complete subgraph) is strictly larger than Bob's largest blue clique; does Bob have a winning strategy for every $n\geq 3$? (Erdős believed the answer is yes.) (2) In the variant where Bob colours two edges after each single edge Alice colours, but Bob must now finish with a strictly larger clique than Alice, does Bob have a winning strategy for every $n>3$? (3) In the game where Alice instead wins if the maximum degree of the red subgraph exceeds the maximum degree of the blue subgraph, who wins? For each variant, determine the winner as a function of $n$, with a strategy.

Acceptance. FULLY RESOLVES (any one of the three games counts, stated per game): a proof determining the winner for all $n$ in the stated range — e.g. an explicit Bob strategy for game (1) valid for every $n\geq 3$ together with a proof that it wins, or a proof that Alice wins for some/all large $n$; and likewise for games (2) and (3). A strategy proof should be complete and, ideally, machine-verifiable. ADVANCES: decide the winner for a range of specific values of $n$ by exhaustive game-tree search, delivered as a reproducible solver plus, for each $n$, the computed game value and a certificate (a winning-strategy tree or a compressed form of it) — extending the set of $n$ with a proven winner beyond what the density results in the background pin down; or prove a new monotonicity/reduction statement strengthening the Malekshahian–Spiro density bounds (e.g. raising the guaranteed density of Bob-winning $n$ for game (1) above $3/4$). Deliver the strategy proof, or the solver plus the table of decided $n$ with winning-strategy certificates.