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One way to approach this question quantitatively is suggested by probability. One can put various measures on the space of all simplicial complexes on $n$ vertices. One perhaps fairly natural measure is to take a random graph and then take the clique complex. This doesn't give us all complexes on $n$ vertices but every complex is homeomorphic to the clique complex of some graph, so we are covering everything up to homeomorphism as $n \to \infty$.

The main point of my paper Topology of random clique complexes is that almost all simplicial complexes arising this way are fairly simple topologically. In particular is shown that for a typical $d$-dimensional clique complex, the homology groups $H_k$ all vanish when $k > \lfloor d/2 \rfloor$ and when $k< d/4$, and that almost all of whatever homology remains is concentrated in the middle dimension $k=\lfloor d/2 \rfloor$.

It is currently an open problem to decide whether the homology is vanishing (or merely small) between $k=d/4$ and $k=d/2$. If one could establish this, then one would be well on the way to showing that almost all flag complexes are homotopy to a wedge of spheres; indeed the last thing to do would be to rule out torsion in middle homology with integer coefficients.

I don't have a good feel for whether either of these things is even true, but I do believe think that this paper gives good anecdotal evidence that most flag complexes are somewhat simple topologically, and is a step in the direction of answering Forman's question. (This particular measure seems especially natural from the point of view of combinatorics, since so many simplicial complexes arise as order complexes of posets, hence are automatically flag complexes.)

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One way to approach this question quantitatively is suggested by probability. One can put various measures on the space of all simplicial complexes on $n$ vertices. One perhaps fairly natural measure is to take a random graph and then take the clique complex. This doesn't give us all complexes on $n$ vertices but every complex is homeomorphic to the clique complex of some graph, so we are covering everything up to homeomorphism as $n \to \infty$.

The main point of my paper Topology of random clique complexes is that almost all simplicial complexes arising this way are fairly simple topologically. In particular is shown that for a typical $d$-dimensional clique complex, the homology groups $H_k$ all vanish when $k > \lfloor d/2 \rfloor$ and when $k< d/4$, and that almost all of whatever homology remains is concentrated in the middle dimension $k=\lfloor d/2 \rfloor$.

It is currently an open problem to decide whether the homology is vanishing (or merely small) between $k=d/4$ and $k=d/2$. If one could establish this, then one would be well on the way to showing that almost all flag complexes are homotopy to a wedge of spheres; indeed the last thing to do would be to rule out torsion in middle homology with integer coefficients.

I don't have a good feel for whether either of these things is even true, but I do believe that this paper gives good anecdotal evidence that most flag complexes are somewhat simple topologically, and is a step in the direction of answering Forman's question. (This particular measure seems especially natural from the point of view of combinatorics, since so many simplicial complexes arise as order complexes of posets, hence are automatically flag complexes.)