F* conjecture

F*.sage

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+# need to find an example that F* is not a subset in the largest hyperplane
+# see https://coauthor.tcsuestc.com/My%20Group/m/mQN2RfiLxYzQ7jAeg for counterexample
+from sage.all import *
+from sage.matroids.all import *
+from sage.graphs.all import *
+
+def check(M):
+    # collect the largest hyperplane and optimal F*
+    Hset=[]
+    Hsize=0
+    Fset=[]
+    Fstrength=1000000
+    for r in range(0,M.rank()):
+        #enumerate all flats
+        rank_deficiency=M.rank()-r
+        for F in M.flats(r):
+            size_deficiency=M.size()-len(F)
+            strength=size_deficiency/rank_deficiency
+            if strength < Fstrength:
+                Fstrength = strength
+                Fset = [F]
+            elif strength == Fstrength:
+                Fset.append(F)
+            if r==M.rank()-1:
+                if Hsize < len(F):
+                    Hsize = len(F)
+                    Hset = [F]
+                elif Hsize == len(F):
+                    Hset.append(F)
+    # print(Hsize, Fstrength)
+    # input()
+    for H in Hset:
+        for F in Fset:
+            if F<=H:
+                return False
+    return True
+    
+
+f = lambda g: g.is_connected()
+for N in range(4,10):
+    #enumerage connect graphs
+    for g in filter(f, graphs(N)):
+        print(g.size())
+        MG=Matroid(g)
+        if check(MG):
+            print("found one example!!")
+            print(g.edges())
+            exit()
+