Simonovits and Sos asked:

Let be a family of graphs with N={1,2,…,n} as the set of vertices. Suppose that every two graphs in the family have a **triangle** in common. How large can be?

(We talked about it in this post.)

One of the most beautiful conjectures in extremal set theory is the Simonovich-Sos Conjecture, the proposed answer to the question above:

Let be a family of graphs with N as the set of vertices. Suppose that every two graphs in the family have a **triangle** in common. Than .

A few weeks ago David Ellis, Yuval Filmus, and Ehud Friedgut proved the conjecture. The paper is now written. The proof uses Discrete Fourier analysis/spectral methods and is quite involved. This is a wonderful achievement.

The example showing that this is tight are all graphs containing a fixed triangle.

Vera Sos’s, a great mathematical hero of mine, is ageless. Nevertheless, she had a birthday conference in early September. The paper by Ellis, Filmus, and Friedgut is dedicated to Vera on the occasion of her birthday. What a nice birthday gift!

Let me add my own wishes: **Happy birthday, Vera!**